# Drift Orbit Bifurcations and Cross-field Transport in the Outer Radiation
Belt: Global MHD and Integrated Test-Particle Simulations

R. T. Desai1 , J. P. Eastwood1, R. B. Horne2, H. J. Allison3, O. Allanson4
C. E. J. Watt5, J. W. B. Eggington1, S. A. Glauert2, N. P. Meredith2, M. O.
Archer1
F. A. Staples6, L. Mejnertsen1, J. K. Tong1, J. P. Chittenden1 corresponding
author<EMAIL_ADDRESS>
###### Abstract

Energetic particle fluxes in the outer magnetosphere present a significant
challenge to modelling efforts as they can vary by orders of magnitude in
response to solar wind driving conditions. In this article, we demonstrate the
ability to propagate test particles through global MHD simulations to a high
level of precision and use this to map the cross-field radial transport
associated with relativistic electrons undergoing drift orbit bifurcations
(DOBs). The simulations predict DOBs primarily occur within an Earth radius of
the magnetopause loss cone and appears significantly different for southward
and northward interplanetary magnetic field orientations. The changes to the
second invariant are shown to manifest as a dropout in particle fluxes with
pitch angles close to 90∘ and indicate DOBs are a cause of butterfly pitch
angle distributions within the night-time sector. The convective electric
field, not included in previous DOB studies, is found to have a significant
effect on the resultant long term transport, and losses to the magnetopause
and atmosphere are identified as a potential method for incorporating DOBs
within Fokker-Planck transport models.

1Blackett Laboratory, Imperial College London, London, UK

2British Antarctic Survey, Cambridge, UK

3GFZ German Research Centre for Geosciences, Potsdam, Germany

5Department of Mathematics, University of Exeter, Penryn/Cornwall Campus; UK

5Department of Mathematics, Physics and Electrical Engineering, Northumbria
University, Newcastle Upon Tyne, UK

6Mullard Space Science Laboratory, University College London, Surrey, UK

##  1 Introduction

The solar wind impinging on the day-side magnetosphere compresses the
magnetospheric field lines and enhances the equatorial magnetic field
magnitude (Mead and Beard, 1964; Northrop, 1966; Roederer, 1970). Close to the
magnetopause, the equatorial field strength can consequently exceed the field
strength at a given particle’s mirror point which causes drift orbits in the
outer magnetosphere to bifurcate as particles become temporarily trapped
within high-latitude pockets of magnetic field minima. Entering and exiting
these non-dipolar regions has been associated with non-conservation of the
particles second adiabatic invariant (Antonova et al., 2003) which, combined
with conservation of the first adiabatic invariant, leads to radial transport
across the magnetic field. It is therefore necessary to account for this
phenomenon in order to understand and predict the dynamics of the outer
radiation belt and its source populations.

The early works of Shabansky and Antonova (1968) and Shabansky (1972)
identified the phenomenon of drift orbit bifurcations (DOBs) and described how
this can affect particles on both open and closed field lines. Early
observations of enhanced energetic proton and electron fluxes in the high-
latitude part of the trapping region were identified by several early
satellite missions (Antonova, 1973; Antonova and Nikolaeva, 1979; Antonova,
1996) which were consistent with later observations of cusp populations (Savin
et al., 1998; Pissarenko et al., 2001; Chen et al., 1997, 1998; Sheldon et
al., 1998). These cusp populations were identified as being produced through
tail injections (e.g. Baker et al., 1998) which drifted around and become
trapped in these high latitude pockets.

The changes to a particles second invariant were first outlined by Antonova et
al. (2003) and quantified through simulations by Öztürk and Wolf (2007) who
numerically integrated particle trajectories within symmetric super-imposed
curl-free magnetic dipoles. These studies demonstrated how the instantaneous
change to a particles pitch angle when undergoing DOBs is attributed to the
particles orbit approaching and intersecting separatrix boundaries within
Hamiltonian phase space (Cary et al., 1986) such that when the equatorial
field intensity exceeds that at the mirror points, particles cross onto a new
orbital contour (Antonova et al., 2003). This conserves the total momentum but
leads to a redistribution of the components parallel and perpendicular to the
magnetic field. The changes to the second invariant were found to be most
significant for particles with small initial values, i.e. particles with pitch
angles close to 90∘. A follow-up study by Wan et al. (2010) looked at DOBs
within asymmetric Tsyganenko (2002) magnetic fields, imposed through a dipole
tilt angle and a non-zero East-West component to the interplanetary magnetic
field.

The long-term evolution of electrons undergoing DOBs was examined by Ukhorskiy
et al. (2011) within Tsyganenko and Sitnov (2007) fields who showed how
particles can remain trapped and meander outward and inward within these non-
dipolar regions for extended time periods. Through comparison with the
magnetic diffusion rates of Brautigam and Albert (2000) they showed how DOBs
can significantly enhance the recirculation and energisation of outer belt
electron populations. Ukhorskiy et al. (2014) subsequently examined DOBs
within Tsyagenko fields subjected to upstream dynamic pressure variations on
ultra-low frequency (ULF) timescales and showed they can lead to large
increases in radial transport rates. The evolution of the distribution
functions in the outer magnetosphere also notably deviated from the diffusion
approximation.

DOBs have been considered in relation to the calculation of the last closed
drift shell (Albert et al., 2018) which presents an important outer boundary
for Fokker-Planck radiation belts models (e.g. Subbotin and Shprits, 2009;
Glauert et al., 2014). These typically use Tsyganenko field models for
calculation of the third invariant as well as the location of the last closed
drift shell but, as the third invariant is undefined for a bifurcating orbit,
there exists a region of the outer magnetosphere where the particle dynamics
are not fully unaccounted for. Global magnetohydrodynamic (MHD) simulation
codes provide an alternate method of modelling the global magnetic field and
test-particle simulations within global MHD fields have yielded valuable
insights into magnetospheric particle dynamics (e,g, Hudson et al., 1997;
Elkington et al., 2002; Kress et al., 2007; Sorathia et al., 2017). The
analysis of DOBs within global MHD fields also allows the inclusion of self-
consistent magnetospheric electric fields not considered in previous studies
of DOBs.

In this study we use the global MHD model, Gorgon (Mejnertsen et al., 2018;
Eggington et al., 2020; Desai et al., 2021), and integrated test-particle
simulations to demonstrate that this modelling approach is capable of
capturing and further studying DOBs. Section 2 first describes the global MHD
simulations and outlines the resolution requirements of $\approx$1/4 RE as
necessary to properly resolve the non-dipolarity in the locally enhanced
magnetic fields near the sub-solar point. Section 3 then describes the test-
particle simulations and how, through a combination of tracing particle
trajectories and magnetic field lines, the simulations are able to conserve
and measure a given particle’s adiabatic invariants to a high level of
precision. Ensembles of energetic electrons are subsequently simulated
traversing the bifurcation region for southward and northward Interplanetary
Magnetic Field (IMF) orientations and the differing transport in each scenario
constrained as a transport map of equatorial pitch angle and radial location.
The simulations agree in predicting large increases in the second invariant
for particles with pitch angles close to 90∘, which is suggested as a
mechanism which produces butterfly pitch angle distributions within the night
sector. The long-term evolution is then examined and the simulations are found
to depend heavily upon the convective electric fields, and DOB-driven losses
to the magnetopause and atmospheric loss cone are suggested as potential
effects which can be incorporated within diffusive radiation belt models.

##  2 Global-MHD Simulations

###  2.1 Method

The resistive MHD simulation code, Gorgon, was developed for the study of
high-energy-density laboratory plasmas (Chittenden et al., 2004; Ciardi et
al., 2007) and has since been adapted to the study of the solar wind-
magnetosphere interaction (Mejnertsen et al., 2016, 2018; Eggington et al.,
2020; Desai et al., 2021). Gorgon is differentiated from other global MHD
codes in that it solves for the vector potential on a staggered grid and
automatically conserves the divergence of the magnetic field, $\nabla\cdot
B=0$, to machine precision without the need for an additional divergence
cleaning operation. Gorgon employs a Message Passing Interface (MPI) domain
decomposition parallelisation and has evolved to include higher order solvers
and a thin shell ionosphere which maps region 1 field aligned currents to and
from the ionosphere (Eggington et al., 2018). The MHD equations are solved on
a cartesian grid extending from -20 to 130 RE in X with the Sun to the left at
–X, and -40 to 40 in Y and Z. In this study we examine the magnetosphere under
the influence of both southward and northward steady solar wind conditions of:
vx = 400 km/s; Ti=Te=5 eV, proton density np=5 cm-3 and with Bz = -2 and 2 nT.
This yields a dynamic pressure of 1.34 nPa and produces a sub-solar
magnetopause stand-off distance of –10.75 and –11.25 RE for the two respective
IMF orientations.

Figure 1: Global magnetic field configurations in the X-Z plane for a grid
resolution of 1/4 RE. Panels (a) and (b) correspond to southward and
northward-oriented IMF conditions, respectively. The colour scale is optimised
to capture high-latitude minima along the outer day-side field lines.

###  2.2 Magnetic Minima

Figure 1 shows a slice through the two simulations run with a grid resolution
of 1/4 RE and zero dipole tilt, two hours after initialisation. The two panels
correspond to southward and northward IMF conditions. The oppositely directed
IMF produce several differences to the configurations of the global fields.
The most obvious of these is that southward IMF induces reconnection at the
sub-solar point as evidenced by field lines being dragged poleward whereas the
northward IMF instead induces reconnection at high latitudes. This effect
slightly erodes the magnetopause such that the sub-solar magnetopause boundary
is located $\approx$0.5 $R_{E}$ further inward for IMF B${}_{z}<$0 and results
in different global convection patterns and a visible near-Earth neutral line
in the magnetotail just beyond X $\approx$ 15 RE. The colour scale is limited
to between 10 and 65 nT to show the variations in the magnetic field strength
along field lines close to the magnetopause. Selected fields lines are traced
in white which reveals the anticipated equatorially enhanced magnetic field
and the minima in the B field at high latitudes. For example, at X = 8 RE in
Figure 1a, the magnetic field is $\approx$20 nT greater at the equator (Z=0)
then the cusps (Z = $\pm$ 4 RE).

Magnetospheric dynamics occur on a variety of scales and even large-scale
magnetospheric phenomena, such as ultra-low-frequency pulsations, have been
shown to be highly dependent on the selected grid resolution (e.g.
Claudepierre et al., 2009). There consequently exists a trade-off between
resolving the phenomenon of interest and a tractable computational problem.
Kress et al. (2007) show DOBs occurring within combined MHD and test particle
simulations using the LFM global MHD model with a non-uniform grid but do not
study this phenomenon further and we therefore first examine increasingly
refined grid resolutions to determine the resolution necessary for resolving
DOBs.

Figure 2 shows the magnetic field strength along selected field lines at 0.5
RE intervals near the magnetopause for three separate simulations with grid
resolutions of 1/2, 1/4 and 1/8 RE respectively, each run with a southward
IMF. In all cases near the magnetopause the magnetic field displays a local
maximum at the equator and two near-symmetric minima on either side of this at
higher latitudes. This local maximum is comparable to those produced within
the semi-empirical Tsyganenko field models (see Wan et al., 2010, Fig. 1
therein). This bifurcation region extends up to $\approx$ 1.5 RE inward from
the magnetopause boundary and appears to diminish in intensity with decreasing
radial distance. This feature appears qualitatively similar at each resolution
but is less pronounced in the lowest resolution case where the simulation
under-estimates the extent of the minima along the outermost field line. This
leaves a region of $\approx$ near the magnetopause where DOBs will not be
accurately capture and due to this, and further test-particle simulations (not
shown), we judge that the 1/4 RE resolution has converged and this resolution
is selected moving forward.

Oscillations in the magnetic field along the outer field-lines are also
evident. These are caused by the southward interplanetary magnetic field (IMF)
inducing reconnection at the subsolar point which means that flux rope
structures are forming in this region. In the 1/8 RE resolution simulations,
the outer-most field lines are also visibly in the process of reconnecting and
being dragged poleward by the convecting solar wind, but this does not mean
the resolutions disagree about the average minima.

Figure 2: Magnetic field strength along selected field lines which map to X =
8.5, 9, 9.5 and 10 RE upstream of the Earth. A dipole field mapping to 8 RE
upstream is shown for comparison.

##  3 Test-Particle Simulations

###  3.1 Method

In this study we trace electron trajectories within the self-consistent MHD
fields using the test-particle approximation. The particles thus see the
magnetic and electric fields but the particles’ currents do not feedback into
the global fields. The MHD simulations utilise a split dipole implementation
(Tanaka, 1994) which allows direct interpolation of the non-dipolar component
to which the analytical curl-free dipolar component is added. We use the
quadratic triangular shape cloud scheme (Hockney and Eastwood, 1981) to map
between the fields and the particles’ location sub-grid and clean spurious
parallel electric fields (Lehe et al., 2009) which are zero within the ideal
MHD approximation. The test-particle simulations are implemented directly into
the Gorgon code via a linked list data structure (Tong et al., 2019) and a
vectorised MPI algorithm balances this additional load across all available
processors. Previous studies have highlighted that in the magnetosphere,
electron trajectories can generally be approximated by the guiding centre
equations of motion (e.g. Elkington et al., 2004) but particles close to the
magnetopause can undergo current sheet scattering in which case finite
gyroradius effects will become significant. We therefore integrate the full
Newton-Lorentz equations of motion using the relativistic Boris integration
scheme (Boris, 1970; Desai et al., 2021). The particles are allowed to orbit
anywhere within a radial sphere of 12 RE and can enter the MHD inner boundary
at 3 RE at which point the MHD equations are no longer solved and the
particles see a dipole field up until an atmospheric loss condition 1,000 km
above the Earth’s surface.

To isolate the effects of drift-orbit-bifurcations from other magnetospheric
phenomena we push the test-particles through individual MHD timesteps. The
fields are therefore static but contain self-consistent electric fields
resulting from plasma convection. We do this for multiple MHD timesteps
obtained during constant solar wind conditions following 2 to 2.5 hours of
initialisation with a geomagnetic dipole moment of M = 7.94$\cdot$1022 A m2.
This therefore removes natural variations associated with magnetopause
reconnection (Russell et al., 1996) and oscillations of the magnetopause
surface (Guo et al., 2010) and any subsequent large-scale oscillations too.

###  3.2 Adiabatic invariants

Constraining particle transport within the simulations requires high precision
determination of a particles’ adiabatic invariants. The first adiabatic
invariant is expressed as,

$\mu=\frac{\gamma m_{0}v_{\perp}}{2B},$ (1)

where $\gamma$ is the relativistic Lorentz factor, m0 the rest mass, and v⟂
the velocity perpendicular to magnetic field, B. This is trivial to calculate
and is available at every location along a particle’s orbit, either within the
simulation or via post-processing. The second adiabatic invariant,

$J=m_{0}\oint\gamma v_{\parallel}\cdot dl,$ (2)

where $v_{\parallel}$ is the velocity parallel to B, is, however, more
complicated to calculate. The discretisation of each particles trajectory into
thousands to millions of steps mean it is not efficient to output every single
datapoint and we therefore evaluate this integral as the particles trajectory
is evolved and output diagnostic data including J/2, at a given particles
mirror point. The third adiabatic invariant is defined as,

$\Phi=\oint_{c}A\cdot dl=-\frac{2\pi B_{0}R^{2}_{E}}{L^{*}},$ (3)

where A is the vector potential. As shown, this is also commonly expressed in
invariant coordinates where B0 is the magnetic field strength at the magnetic
equator on the Earth’s surface, and L∗ is the particles L shell were all non-
dipolar components slowly turned off. The third adiabatic invariant expressed
in Equation 3 is, however, undefined for a bifurcating drift shell (e.g.
Öztürk and Wolf, 2007; Ukhorskiy et al., 2011) which is problematic for
calculating radial transport rates. We therefore calculate radial transport
rates in terms of dipole L shell under the assumption $\Delta$ L $\approx$
$\Delta$ L∗ at a given location in the magnetosphere and leave the problem of
converting between physical and invariant coordinate systems for a further
dedicated endeavour.

Despite this assumption, measuring radial changes is not straight-forward as L
has to be measured at a common location, before and after a drift orbit and a
particle will in all probabilities arrive at the same local time with a
different bounce and gyrophase. To account for this we offset the particle’s
gyroradius, and trace the gyro-centre location along the magnetic field down
to the magnetic equator for both the initial and final positions.

Figure 3: Three MeV electron trajectories undergoing (a) a stable drift orbit
at L=6, (b) a bifurcated orbit which leads to an increase to the particles
pitch angle and (c) a bifurcated orbit which leads to a decrease to the
particles pitch angle. Lower panels (d–f) show the corresponding first and
second adiabatic invariants and dipole L shell where the legends denote the
change across the entire drift orbit.

###  3.3 Drift Orbit Bifurcations

Figure 3a–c shows a selection of three 1 MeV electron orbits which represent a
stable equatorially mirroring particle and two particles undergoing DOBs.
Figure 3a shows the full orbit of the first electron which has an equatorial
start position of L = 6 RE downtail and initial equatorial pitch angle,
$\alpha_{0}$ = 45∘. The electron orbits anti-clockwise in MLT and mirrors to
slightly higher latitudes on the day-side due to the compression of the
magnetosphere before returning to its initial location. Figure 3b shows the
second electron initialised at L = 8 RE downtail with $\alpha_{0}$ = 90 ∘.
This electron drifts along the magnetic equator into the bifurcation region on
the day side and is deflected away from the local maximum at around 10.30 MLT
and becomes trapped beneath the equator. The pitch angle increases
significantly as it starts mirroring within the minimum B pocket in the
southern hemisphere and, upon exiting this region near 13:30 MLT, the pitch
angle increases further and the electron once again mirrors about the magnetic
equator as it drifts back to the tail. Figure 3c shows the third electron
starting at approximately the same radial distance as the second electron but
this time with $\alpha_{0}$ = 65 ∘. This third electron similarly avoids the
local equatorial maximum on the day-side but this time its pitch angle
decreases as the particle becomes trapped within the minimum B pocket and
decreases even further upon exiting.

Figure 3d-e displays the corresponding first and second invariants for these
particle trajectories together with the particles L shell to represent radial
transport. In all cases the first invariant remains approximately constant
with a overall change from start to end of less than 1 %. In the case of the
stably trapped particle shown in Figure 3a, a sinusoidal variation is apparent
due to the particle’s interaction with the dawn-dusk electric field. This
effect is also visible in the two bifurcating particle orbits shown in Figure
3b-c, although this field displays some modulation at noon close to the
magnetopause boundary.

The second invariant is well-conserved for the stably trapped electron but
undergoes several changes for the bifurcating orbits. The changes to the pitch
angles noted in Figures 3b–c, correspond to large step changes in J. This is
also not constant within the trapped region and undergoes further variations
which are attributed to the use of dynamic MHD fields and realistic local
variations and inhomogeneities. The second invariant changes, or lack thereof,
are mirrored in the electrons’ radial position due to the local conservation
of the first invariant. The stably trapped electron consequently returns to a
radial position less than 0.006 RE from its initial position. This small
discrepancy could be attributed to the presence of a non-zero electric field
but also represents the limit of our accuracy in measuring radial transport.
The bifurcating drift orbits, however, do not return to the same radial
position. Figure 3f shows how the two instantaneous increases to the second
invariant corresponds to net outward radial transport of approximately 1/4 RE
and how a decrease in the second invariant corresponds to net inward radial
transport of the same amount. The L shell parameters vary significantly along
the drift orbit due to the compression of the global field and further
highlights why we measure radial transport at a common MLT.

It should be noted that in the DOB trajectories shown, the electrons either
exhibit two increases or two decreases in J. This is not always the case and
some particles exhibit an increase followed by a decrease in J, or vice versa,
upon entering and exiting the minimum B pockets. This predominantly occurs for
intermediary pitch angles, as discussed in the preceding Section 3.4, and is
dependent upon the bounce phase which controls the location at which the
particle enters the minimum B pocket.

###  3.4 Transport Maps

To further constrain this radial transport mechanism, we derive maps for an
ensemble of 1 MeV electrons initialised in the tail. The distribution covers a
radial range of L = 6–10 in steps of 0.05, pitch angles, $\alpha_{0}$ =
30–150∘ in steps of 1∘, and an MLT range of 0:00 – 00:02 minutes in steps of
10 seconds to resolve the electron bounce-drift period. The particles are
traced through the magnetosphere for both southward and northward IMF
conditions where the dynamic pressure is held constant, in order to examine
effects associated with the differing structures of the reconnecting
magnetopause boundary. The electrons are traced through a single drift orbit,
as displayed in Figure 3.

Figure 4: Transport maps for southward IMF conditions. (a) shows
$\Delta\bar{J}$ and (b) shows $\Delta\bar{L}$, both as a function of
equatorial pitch angle $\alpha_{0}$ and initial position r0 which is
equivalent to dipole L shell. Lower panels (c) and (d) show the standard
deviation, $\sigma$, over the multiple bounce phases.

Figures 4 and 5 show the results of these two simulations for southward and
northward IMF orientations, respectively. The mean change of the ensemble,
$\langle J\rangle$ and $\langle L\rangle$, are shown together with the
standard deviation, $\sigma$, of electrons with different bounce phases. At
larger L shells, the resultant transport maps show a drop-out in electrons
which encounter the magnetopause loss cone which is higher for particles
mirroring further from the equator. For large regions of the maps at lower L
shells, $\langle J\rangle$ and $\langle L\rangle$ are approximately zero which
represents negligible transport and stably trapped electrons analogous to
those shown in Figure 3a. Close to the magnetopause loss cone, however, a
large region is evident where particles exhibit significant transport
characteristic of the orbits shown in Figure 3b and 3c.

For pitch angles at or close to 90 ∘, the electrons exhibit large increases to
both $\langle J\rangle$ and $\langle L\rangle$ whereas particles outside this
region typically exhibit decreases to $\langle J\rangle$ and $\langle
L\rangle$. This trend appears symmetric about 90 ∘, as anticipated for the
symmetric simulation conditions and possesses the same shape as the
magnetopause loss cone boundary. The standard deviation across the bounce
phase is in some instances comparable to the transport itself which highlights
further complexities in trying to represent DOB transport in invariant space.
In Figures 4 and 5, $\sigma$ is largest for larger pitch angles, most notably
in Figure 4 along two bands which run parallel to the loss cone. These
represent regions where $\langle J\rangle$ and $\langle L\rangle$ can either
increase or decrease, depending on the point in the electrons bounce phase
that the drift shell bifurcates. There is also a notable minimum in the
transport between 1.6 and 8 RE near 90 ∘ for southward IMF.

Several differences are apparent between Figures 4 and 5. The magnetopause
loss cone is at higher L shells for the Northward IMF conditions due to the
different shape and size of the dayside magnetopause. In both Figures,
$\langle\Delta J\rangle$ and $\langle\Delta L\rangle$ peak inside the
magnetopause boundary but for southward IMF, this effect is more pronounced
and the red regions denoting increases to J and L, reflect the shape of the
magnetopause loss cone. Regions of large transport are also present directly
adjacent to the loss cone which show the effects of current sheet scattering
which can also cause particles to enter the atmospheric loss cone.

Figure 5: Transport maps for northward IMF conditions. (a) shows
$\langle\Delta J\rangle$ and (b) shows $\langle\Delta L\rangle$, both as a
function of equatorial pitch angle $\alpha_{0}$ and L shell. Lower panels (c)
and (d) show the standard deviation, $\sigma$, over the bounce phase
distribution.

###  3.5 Observable Signature

It is worth noting that the effects of DOBs are not easy to constrain via
observations. The simulations predict a local depletion near the subsolar
point of particles with small pitch angles and enhanced fluxes at higher
latitudes, the latter of which indeed has been identified in several studies
(Antonova, 1973; Antonova and Nikolaeva, 1979; Antonova, 1996; Savin et al.,
1998; Pissarenko et al., 2001; Chen et al., 1997, 1998; Sheldon et al., 1998).
Flux in the outer magnetosphere can, however, vary by orders of magnitude and
this inherent variability makes it difficult to identify DOB transport
predicted via theory and numerical simulations.

Figure 6 shows the equatorial pitch angle distribution on the night side
corresponding to the maps shown in Figures 4 and 5. This shows that the
consequence of large jumps in J for particles with small initial values of J,
is net transport away from $\alpha_{0}$ = 90 degrees. Butterfly PADs are
observed throughout the night-time sector (West et al., 1973) and have been
identified as deriving from drift shell splitting combined with magnetopause
shadowing (e.g. Klida and Fritz, 2013), interactions with ULF waves (e.g.
Kamiya et al., 2018) and particle injections (e.g. Artemyev et al., 2015).
Here we identify DOBs as a mechanism which might also produce such a
distribution. It is important to note, however, that the test-particle
simulations presented herein do not contain kinetic effects such as the self-
consistent generation and interaction with plasma instabilities. McCollough et
al. (2012) suggest that anisotropies can arise purely due to DOBs and these,
as well as further plasma waves observed within the magnetosphere, might act
to scatter particles and the dropouts near 90∘ therefore might be less
pronounced than appear in Figure 6.

Figure 6: Butterfly pitch angle distribution after 1 drift orbit for (a)
southward and (b) northward IMF conditions .

###  3.6 Radial Evolution

In this section we examine the long term transport associated with the
invariant changes displayed in Figures 4 and 5. The transport rates of
relativistic electrons in the radiation belts are typically approximated by
the Fokker-Plank equation where the second moment of the distribution function
scales with time as $\langle(\Delta L)^{2}\rangle=2D_{LL}t$, where $D_{LL}$ is
a diffusion coefficient which is applied in a drift averaged sense. This
description is instantaneously utilised at multiple locations within the
magnetosphere, such that a radial diffusion coefficient represents a Dirac
delta function of particles at a particular radial distance, summed over
azimuth. Within our simulations we see significant loss to the magnetopause
over time which renders the resultant transport characteristics of an ensemble
unphysical, and further complicates the scenario where the azimuthal third
invariant is already undefined. Ukhorskiy et al. (2014) examine the evolution
of an ensemble with near 90 ∘ pitch angles at a specific radial distance to
examine the diffusion characteristics. Here, to study the effect of DOBs on
outer belt electrons, we choose to analyse the bulk transport characteristics
of a distribution function which includes multiple L shells, to examine and
isolate the net effect of DOBs on the outer radiation belt near the
magnetopause.

Figure 7 shows the radial evolution of the ensemble which encompasses all
pitch angles and bounce phases. The three different times correspond to after
1 drift orbit, after 3 and then 6 hours where the particles have eventually
undergone 25 or more drift orbits depending on their L shell. The initially
flat distribution functions exhibits an initial decrease at higher L shells
following a single drift orbit which corresponds to the loss cones shown in
Figure 4 and 5. The distribution is also no longer flat and features several
peaks and troughs. Following this, further losses occur over time, most
notably for southward IMF conditions. Figure 8 shows further characteristics
of the radial distribution function over time, starting with the distributions
shown in Figures 4 and 5. Also shown are the same particle simulations run
with the electric field set to zero in order to constrain the transport
characteristics associated purely with the non-dipolar magnetic structure of
the global fields. This therefore removes the effect of the electric fields
and allows comparison with prior results obtained from empirical and semi-
empirical magnetospheric fields (Öztürk and Wolf, 2007; Wan et al., 2010;
Ukhorskiy et al., 2011, 2014).

Figure 7: Summed distribution function at three instances over 6 hours. Upper
panels (a–c) shows the evolution under southward IMF conditions whereas lower
panels (d–f) shows the evolution under northward IMF conditions.

We first concentrate on the scenarios where we include the full magnetospheric
magnetic and electric fields, shown by the solid lines. In Figure 8a, for
southward IMF, the mean L shell of the entire distribution, $\langle
L\rangle$, initially moves inward due to further initial losses to the
magnetopause on the drift orbits following the initial one and then slowly
moves outward over the next 6 hours. For northward IMF, the distribution does
not encounter the same immediate losses but also moves steadily outward. This
trend is found during multiple MHD timesteps and is thus attributed to the
convective electric field as opposed to a $\delta B$ component induced by
ultra-low-frequency fluctuations. Over this time period Figure 8b shows the
squared radial change, $\langle(\Delta L)^{2}\rangle$. For the southward IMF
conditions, the magnetopause losses dominate over the first half of the
simulations and produce decreases of this parameter up to 3 hours after which
this then exponentially increases. For northward IMF conditions this
exponential increase is immediately apparent. Figure 8c shows that during this
time period losses to the magnetopause are significant but more so for the
southward IMF conditions where nearly half of the particles are lost after 6
hours.

These scenarios display notable differences when the electric field is
excluded, as shown by the dashed lines. In these scenarios, Figure 8a shows
the average of the distributions maintain a near-constant value, although for
southward IMF there is a small decline in this parameter. For northward IMF,
$\langle\Delta L\rangle$ is less than 10-4 but for the southward case this
initially rises sharply which is followed by an approximately linear increase
over the next 5.5 hours. This is mirrored in the magnetopause losses which
appear minimal for northward IMF but are more pronounced for southward IMF
where over 20% of the particles are eventually lost. The northward IMF
scenario, where the radial distribution stays approximately constant and does
not diffuse over time, is in qualitative agreement with previous studies (e.g.
Ukhorskiy et al., 2011), where particles can undergo DOBs for extended time
periods and traverse back and forth in the magnetosphere as their second
invariant and radial locations increase and decrease. There is a small net
outward trend, however, and for southward IMF the simulations clearly
demonstrate that the net effect of DOBs is outward radial transport and
continual loss to the magnetopause.

When the particles see both B and E, the evolution, as shown, appears super-
diffusive, i.e. $(\Delta L)^{2}\propto t^{a}$ where a$>1$, and the discussion
of the evolution in terms of diffusive behaviour is also problematic due to
the finite and varying number of particles in the simulation and the
aforementioned large influence of the convective electric fields. There is
currently significant interest in understanding the limits of the diffusive
regime (e.g. Omura et al., 2009; Allanson et al., 2020; Trotta et al., 2020)
where the advection of ensembles of particles, not included in traditional
derivations of the Fokker-Planck equation, take hold. The self-consistent
global MHD and test particles simulations appear to indicate that convective
and advective effects are significant for DOBs but also that there appears to
be some underlying diffusive behaviour associated with particles encountering
this non-dipolarity in the outer magnetosphere, as shown when the E field is
set to zero for southward IMF and the consistent magnetopause losses.

Figure 8: (a) shows the mean location of the distribution function over 6
hours, (b) shows the corresponding mean change squared, $\langle(\Delta
L)\rangle^{2}$ and (c) shows number of particles remaining in the simulations
normalised to the initial number following the initial drift orbit.

##  4 Discussion and Conclusions

In this study we have developed test particle simulation integrated within
Gorgon global MHD simulations and demonstrated that we can capture DOB
dynamics of the outer radiation belt. Through a combination of particle
trajectory and magnetic field line tracing, we were able to measure particle
transport to a high level of precision and used this to constrain radial
transport resulting from DOBs. The overall transport characteristics broadly
agree with previous studies using semi-empirical Tsyganenko fields that this
phenomenon affects particles within an Earth radius of the magnetopause loss
cone and can induce outward or inward transport. Several complexities were,
however, revealed within this region due to the dynamic nature of the MHD
simulations and inclusion of self-consistent electric fields.

The MHD simulations reproduced an equatorial field maximum comparable to semi-
empirical Tsyganenko magnetic field models and preliminary convergence tests
determined that a grid resolution of 1/4 RE was sufficient to resolve this
magnetic structure. Through the analysis of individual trajectories, the
increase or decrease to an electron’s pitch angle when entering and exiting
the resultant high latitude pocket of magnetic field minimum, was shown to
directly correlate with an increase or decrease in radial distance. This
transport was then further constrained for ensembles of electrons for both
southward and northward IMF orientations and the resultant transport was
mapped as a function of pitch angle and L shell. The instantaneous transport
rates displayed notable variations within this region with the most
significant transport observed for electrons with small initial values of the
second invariant occurring away from the magnetopause surface. This was
surprising as the equatorial maximum in the magnetic field appeared strongest
closest to the magnetopause boundary. The transport maps obtained for
southward IMF conditions also displayed notable structure with two distinct
bands of increased transport occurring inward from the magnetopause surface
and running parallel to the pitch angle dependent magnetopause loss cone.
These differences represent complexities to DOBs which aren’t apparent from
initial considerations of the global field topology.

The long term transport resulting from DOBs appeared to be strongly dependent
on the convective electric fields contained within the global fields, for both
southward and northward IMF conditions. Including the electric fields appeared
to induce net outward transport of particles orbiting within an Earth radius
of the magnetopause loss cone. Several phenomena are capable of inducing
dynamics effects close to the magnetopause boundary, even during steady solar
wind conditions, such as local plasma motions near the magnetopause resulting
from reconnection (Eastwood et al., 2015), or Kelvin-Helmholtz surface waves
(Guo et al., 2010). The resultant transport consequently did not follow the
diffusion approximation where $(\Delta$L)2 $\propto$ t. To further understand
this we excluded the electric field to examine the effects of the particles
interacting purely with the magnetic fields. These results also displayed
differences between southward and northward IMF conditions and the southward
IMF conditions notably induced transport which did scale as $(\Delta$L$)^{2}$
$\propto$ t and indicates an underlying diffusive aspect to this phenomenon.

Predicting radiation belt dynamics using diffusive models typically means
competing effects have to be treated individually (Glauert and Horne, 2005;
Subbotin and Shprits, 2009). The simulations reported herein confirms the
difficulties in incorporating DOBs into this approach, namely due to the non-
diffusive nature of the transport and the large variations in the transport as
a function of the electrons bounce phase. The long-term transport did,
however, display significant losses to the magnetopause which remained
reasonably constant for all scenarios examined. This net loss to the
magnetopause and atmospheric loss cones highlights that the idea of a last-
closed-drift-shell is more complex than an instantaneous location coinciding
with the magnetopause loss cone (Albert et al., 2018) but also presents a
potential method for incorporating the net effects of DOBs into Fokker-Planck-
type models via a loss term acting near the magnetopause.

It was noted that despite observations of trapped populations in the high-
latitude cusps (Antonova, 1973; Antonova and Nikolaeva, 1979; Antonova, 1996;
Savin et al., 1998; Pissarenko et al., 2001; Chen et al., 1997, 1998; Sheldon
et al., 1998), the effects of DOBs are not easy to constrain via observations
and it remains unclear to what extent DOBs persist and influence the highly
variable outer radiation belt. We therefore identify butterfly pitch angle
distributions in the night-time sector as a further observational signature
which may be observed and linked to DOBs.

## Acknowledgements

RTD, JPE and JPC acknowledge funding from NERC grant NE/P017347/1 (Rad-Sat).
HJA is supported by an Alexander Von Humboldt fellowship. OA is supported by a
NERC Independent Research Fellowship NE/V013963/1. CEJW acknowledges grants
NE/V002759/2 and ST/W002078/1. MOA holds a UKRI (STFC / EPSRC) Stephen Hawking
Fellowship EP/T01735X/1. JWBE is supported by Grant NE/P017142/1 (SWIGS). RH,
SG and NM would like to acknowledge the Natural Environment Research Council
Highlight Topic grant NE/P10738X/1 (Rad-Sat) and the NERC grants NE/V00249X/1
298 (Sat-Risk) and NE/R016038/1. This work used the Imperial College High
Performance Computing Service (doi: 10.14469/hpc/2232).

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# Some new results on functions in $C(X)$ having their support on ideals of
closed sets

Sagarmoy Bag  Department of Pure Mathematics, University of Calcutta, 35,
Ballygunge Circular Road, Kolkata 700019, West Bengal, India
<EMAIL_ADDRESS>,  Sudip Kumar Acharyya  Department of Pure
Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata
700019, West Bengal, India<EMAIL_ADDRESS>,  Pritam Rooj  NIIT
University, Neemrana, Rajasthan, PIN 301705, India
<EMAIL_ADDRESS>and  Goutam Bhunia  Department of Pure
Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata
700019, West Bengal, India<EMAIL_ADDRESS>
###### Abstract.

For any ideal $\mathcal{P}$ of closed sets in $X$, let $C_{\mathcal{P}}(X)$ be
the family of those functions in $C(X)$ whose support lie on $\mathcal{P}$.
Further let $C^{\mathcal{P}}_{\infty}(X)$ contain precisely those functions
$f$ in $C(X)$ for which for each $\epsilon>0,\\{x\in X:\lvert
f(x)\rvert\geq\epsilon\\}$ is a member of $\mathcal{P}$. Let
$\upsilon_{C_{\mathcal{P}}}X$ stand for the set of all those points $p$ in
$\beta X$ at which the stone extension $f^{*}$ for each $f$ in
$C_{\mathcal{P}}(X)$ is real valued. We show that each realcompact space lying
between $X$ and $\beta X$ is of the form $\upsilon_{C_{\mathcal{P}}}X$ if and
only if $X$ is pseudocompact. We find out conditions under which an arbitrary
product of spaces of the form locally-$\mathcal{P}/$ almost
locally-$\mathcal{P}$, becomes a space of the same form. We further show that
$C_{\mathcal{P}}(X)$ is a free ideal ( essential ideal ) of $C(X)$ if and only
if $C^{\mathcal{P}}_{\infty}(X)$ is a free ideal ( respectively essential
ideal ) of $C^{*}(X)+C^{\mathcal{P}}_{\infty}(X)$ when and only when $X$ is
locally-$\mathcal{P}$ ( almost locally-$\mathcal{P}$). We address the problem,
when does $C_{\mathcal{P}}(X)/C^{\mathcal{P}}_{\infty}(X)$ become identical to
the socle of the ring $C(X)$. Finally we observe that the ideals of the form
$C_{\mathcal{P}}(X)$ of $C(X)$ are no other than the $z^{\circ}$-ideals of
$C(X)$.

###### Key words and phrases:

Compact support, pseudocompact space, intermediate ring, pseudocompact
support, essential ideal, $z^{\circ}$-ideal, socle, $C$-type ring.

###### 2010 Mathematics Subject Classification:

Primary 54C40; Secondary 46E25

The first author thanks the NBHM, Mumbai-400 001, India, for financial support

The fourth author thanks the NBHM, Mumbai-400 001, India, for financial
support

##  1\. Introduction

In what follows $C(X)$ stands for the ring of all real valued continuous
functions on a completely regular Hausdorff topological space $X$. $C^{*}(X)$,
as usual denotes the subring of $C(X)$ containing those functions, which are
bounded over $X$. Let $\mathcal{P}$ be a family of closed subsets of $X$
satisfying the following two conditions:

  1. (1)

If $A,B\in\mathcal{P}$ then $A\cup B\in\mathcal{P}$.

  2. (2)

If $A\in\mathcal{P}$ and $B\subseteq A$ with $B$ closed in $X$ then
$B\in\mathcal{P}$

i.e. $\mathcal{P}$ is an ideal of closed sets in $X$. For any $f\in C(X)$,
$Z(f)=\\{x\in X:f(x)=0\\}$ stands for the zero set of $f$. Let $\Omega(X)$
denote the set of all ideals of closed sets in $X$. For each
$\mathcal{P}\in\Omega(X)$ set $C_{\mathcal{P}}(X)=\\{f\in
C(X):\operatorname{cl}_{X}(X-Z(f))\in\mathcal{P}\\}$ and
$C^{\mathcal{P}}_{\infty}(X)=\\{f\in C(X):\\{x\in X:\lvert
f(x)\rvert\geq\epsilon\\}\in\mathcal{P}\text{ for each }\epsilon>0\\}$. If
$\mathcal{P}$ is chosen to be the family of compact subsets of $X$, then
$C_{\mathcal{P}}(X)$ and $C^{\mathcal{P}}_{\infty}(X)$ coincide with the ring
$C_{K}(X)$ of real valued continuous functions on $X$ with compact support and
the ring $C_{\infty}(X)$ of real valued continuous functions which vanish at
infinity respectively. These two rings $C_{\mathcal{P}}(X)$ and
$C_{\mathcal{P}}^{\infty}(X)$ have been introduced first time in 2010 in [2].
However the same pair of rings have been reintroduced by following some dual
techniques in 2015 in [20].

As usual $\beta X$ stands for the Stone-$\check{C}$ech compactification of
$X$. For any $f\in C(X),f^{*}:\beta X\mapsto\mathbb{R}\cup\\{\infty\\}$ is its
unique continuous extension over $\beta X$ into the one-point compactification
of $\mathbb{R}$. For $\mathcal{P}\in\Omega(X)$, let
$\upsilon_{C_{\mathcal{P}}}X=\\{p\in\beta X:f^{*}(p)\in\mathbb{R}\text{ for
all }f\in C_{\mathcal{P}}(X)\\}$. It is easy to see that
$\upsilon_{C_{\mathcal{P}}}X$ is a real compact space lying between $X$ and
$\beta X$. The first main theorem of this article states that each real
compact space between $X$ and $\beta X$ could be achieved as a space of the
form $\upsilon_{C_{\mathcal{P}}}X$ for some $\mathcal{P}\in\Omega(X)$ if and
only if $X$ is pseudocompact (Theorem 2.3).

For any $\mathcal{P}\in\Omega(X)$, we call $X$ locally-$\mathcal{P}$ if each
point $x$ of $X$ has an open neighbourhood whose closure in $X$ belongs to
$\mathcal{P}$. More generally $X$ is called almost locally-$\mathcal{P}$ if
each nonempty open set $U$ in $X$ contains a nonempty open set $V$ such that
$\operatorname{cl}_{X}V$ is a member of $\mathcal{P}$. If $\mathcal{P}$ is the
ideal of all compact sets in $X$, then locally-$\mathcal{P}$ spaces and almost
locally-$\mathcal{P}$ spaces bog down to locally compact spaces and almost
locally compact spaces respectively. The later class of spaces is introduced
in [4]. We have introduced the notion product of ideals of closed sets in the
following manner: If $\\{X_{\alpha}:\alpha\in\Lambda\\}$ is an indexed family
of topological spaces with $\mathcal{P}_{\alpha}\in\Omega(X_{\alpha})$ for
each $\alpha\in\Lambda$, then $\prod_{\alpha\in\Lambda}\mathcal{P}_{\alpha}$
is defined to be the smallest member of
$\Omega(\prod_{\alpha\in\Lambda}X_{\alpha})$ containing sets of the form
$\prod_{\alpha\in\Lambda}A_{\alpha}$, where $A_{\alpha}$ belongs to
$\mathcal{P}_{\alpha}$ for each $\alpha$ in $\Lambda$. We have shown that the
product space $\prod_{\alpha\in\Lambda}X_{\alpha}$ is
locally-$\prod_{\alpha\in\Lambda}\mathcal{P}_{\alpha}$ if and only if
$X_{\alpha}$ is locally-$\mathcal{P}_{\alpha}$ for each $\alpha\in\Lambda$ and
$X_{\alpha}\in\mathcal{P}_{\alpha}$ for all but finitely many $\alpha$’s in
$\Lambda$. We have further checked that $\prod_{\alpha\in\Lambda}X_{\alpha}$
is almost locally-$\prod_{\alpha\in\Lambda}\mathcal{P}_{\alpha}$ when and only
when $X_{\alpha}$ is almost locally-$\mathcal{P}_{\alpha}$ for each $\alpha$’s
in $\Lambda$ and $X_{\alpha}\in\mathcal{P}_{\alpha}$ for all but finitely many
$\alpha$’s in$\lambda$. For the particular choice of $\mathcal{P}_{\alpha}$ as
the ideal of all compact sets in $X_{\alpha}$ for each $\alpha$ in $\Lambda$,
the first result reads: $\prod_{\alpha\in\Lambda}X_{\alpha}$ is locally
compact if and only if each $X_{\alpha}$ is locally compact and all but
finitely many $X_{\alpha}$’s are compact, a well known result in general
topology. For the same choice of $\mathcal{P}_{\alpha}$’s, the second result
says that $\prod_{\alpha\in\Lambda}X_{\alpha}$ is almost locally compact if
and only if $X_{\alpha}$ is almost locally compact for each $\alpha$ in
$\Lambda$ and $X_{\alpha}$’s are compact for all but finitely many $\alpha$’s
in $\Lambda$. This later fact is established in [4]. Furthermore on choosing
$\mathcal{P}_{\alpha}$ as ideal of closed bounded sets in $X_{\alpha}$we have
established analogous results involving locally pseudocompact spaces and
almost locally pseudocompact spaces. It is easy to check that
$C_{\mathcal{P}}(X)\subseteq C^{\mathcal{P}}_{\infty}(X)$ with
$C_{\mathcal{P}}(X)$ an ideal of the ring $C(X)$ and
$C^{\mathcal{P}}_{\infty}(X)$ a subring of $C(X)$. For some specific choice of
$\mathcal{P}$, say $\mathcal{P}\equiv$ the ideal of all compact sets in
$X\equiv K$ say, $C^{\mathcal{P}}_{\infty}(X)=C_{\infty}(X)\subseteq
C^{*}(X)$. In general however $C^{\mathcal{P}}_{\infty}(X)$ may not be
contained in $C^{*}(X)$, this is easily seen on taking $X=\mathbb{R}$ and
$\mathcal{P}$, the ideal of all closed Lindelöf subsets of $X$. But if $f\in
C^{\mathcal{P}}_{\infty}(X)$ and $g\in C^{*}(X)$, then it follows that $fg\in
C^{\mathcal{P}}_{\infty}(X)$. Hence $C^{\mathcal{P}}_{\infty}(X)$ is an ideal
of the ring $C^{\mathcal{P}}_{\infty}(X)+C^{*}(X)\equiv\\{f+g:f\in
C^{\mathcal{P}}_{\infty}(X),g\in C^{*}(X)\\}$. We have shown that
$C_{\mathcal{P}}(X)$ is a free ideal of $C(X)$ if and only if
$C^{\mathcal{P}}_{\infty}(X)$ is a free ideal of
$C^{\mathcal{P}}_{\infty}(X)+C^{*}(X)$ if and only if $X$ is
locally-$\mathcal{P}$. We recall that an ideal $I$ in $C(X)$ is free if
$\cap_{f\in I}Z(f)=\phi$. We have established that for
$\mathcal{P}\in\Omega(X)$, $C_{\mathcal{P}}(X)$ is an essential ideal of
$C(X)$ when and only when $C^{\mathcal{P}}_{\infty}(X)$ is an essential ideal
of $C^{\mathcal{P}}_{\infty}(X)+C^{*}(X)$ and this is precisely the case if
and only if $X$ is almost locally-$\mathcal{P}$. With $\mathcal{P}\equiv K$,
this reads: $C_{K}(X)$ is an essential ideal of $C(X)$ if and only if
$C_{\infty}(X)$ is an essential ideal of $C^{*}(X)$ when and only when $X$ is
almost locally compact. This last result was established in [4]. Incidentally
a nonzero ideal $I$ in a commutative ring $R$ (with or without identity) is
called an essential ideal if it intersects each nonzero ideal in $R$
nontrivially, this means that for each $a\neq 0$ in $R$, there exists $b\neq
0$ in $R$ such that $ab\neq 0$ and $ab\in I$. We have recorded a second
special case of our general result involving $C_{\mathcal{P}}(X)$ and
$C^{\mathcal{P}}_{\infty}(X)$, mentioned above in the following manner: the
ring $C_{\psi}(X)$ of all functions in $C(X)$ with pseudocompact support is an
essential ideal of $C(X)$ if and only if the ring
$C^{\psi}_{\infty}(X)\equiv\\{f\in C(X):\text{ for each }\epsilon>0\text{ in
}\mathbb{R},\operatorname{cl}_{X}\\{x\in X:\lvert f(x)\rvert>\epsilon\\}\text{
is pseudocompact}\\}$ is an essential ideal of $C^{*}(X)$ when and only when
$X$ is almost locally pseudocompact meaning that each nonempty open set $U$ in
$X$ contains nonempty open set $V$ such that $\operatorname{cl}_{X}V$ is
pseudocompact. The ring $C^{\psi}_{\infty}(X)$ may be called the pseudocompact
analogue of the ring $C_{\infty}(X)$. As a follow up of the results of this
type, we have addressed the problem: when does $C_{\mathcal{P}}(X)$
(respectively $C^{\mathcal{P}}_{\infty}(X)$) become identical to the
intersection of all essential ideals of $C(X)$ also known as the socle of
$C(X)$. We conclude this article after establishing that ideals of the ring
$C(X)$ of the form $C_{\mathcal{P}}(X)$ with $\mathcal{P}\in\Omega(X)$ are
precisely the $z^{\circ}$-ideals of $C(X)$, investigated amongst others by
Azarpanah, Karamzadeh, Aliabad and Karavan in a series of articles
[6],[7],[8],[9]. On realizing $z^{\circ}$-ideals of $C(X)$ as ideals of the
type $C_{\mathcal{P}}(X)$, we have given an alternative short proof of the
fact that each ideal of $C(X)$ consisting only of zero divisors extends to a
$z^{\circ}$-ideal of $C(X)$. For more information about the rings
$C_{\mathcal{P}}(X)$ and $C^{\mathcal{P}}_{\infty}(X)$, we refer to the
articles [2], [3]. For undefined terms see [12].

##  2\. A Characterization of pseudocompact spaces $X$ in terms of ideals of
the form $C_{\mathcal{P}}(X)$

For a subset $A$ of $C(X)$ and a subset $T$ of $\beta X$, we will use the
notation $\upsilon_{A}X=\\{p\in\beta X:f^{*}(p)\in\mathbb{R}\text{ for each
}f\in A\\}$ and $C_{T}=\\{f\in C(X):f^{*}(p)\in\mathbb{R}\text{ for each }p\in
T\\}$. Clearly $\upsilon_{C}X=\upsilon X$, the Hewitt real compactification of
$X$ and $\upsilon_{C^{*}}X=\beta X$. A ring is said to be of $C$-type if it is
isomorphic to a ring of the form $C(X)$ for some space $X$. By an intermediate
ring we mean a ring that lies between $C^{*}(X)$ and $C(X)$. Let $\Sigma(X)$
stand for the set of all intermediate rings. For $A,B\in\Sigma(X)$ we write
$A\sim B$ if and only if $\upsilon_{A}X=\upsilon_{B}X$.
${}^{\prime}\sim^{\prime}$ is an equivalence relation on $\Sigma(X)$ causing a
partition of this set into disjoint equivalence classes. Each equivalence
class has a largest member, the largest member of the equivalence class of
$A(X)$ is given by $\\{g\mid_{X}:g\in C(\upsilon_{A}X)\\}$, which contains
precisely those functions in $C(X)$ which have continuous extension over the
$A$-compactification $\upsilon_{A}X$of $X$. Incidentally $C^{*}(X)$ is the
lone member of its own equivalence class. It is easy to see that for any
subset $T$ of $\beta X$, $C_{T}$ is an intermediate ring. The following fact
characterizing $C$-type rings amongst members of $\Sigma(X)$ is established in
[10]:

######  Theorem 2.1.

For a ring $A(X)\in\Omega(X)$, the following three statements are equivalent:

  1. (1)

$A(X)$ is a $C$-type ring.

  2. (2)

$A(X)$ is the largest member of its own equivalence class modulo the relation
${}^{\prime}\sim^{\prime}$ on $\Sigma(X)$.

  3. (3)

There exists a subset $T$ of $\beta X$ such that such that $A(X)=C_{T}$.

We also write down the following proposition giving some natural examples of
$C$-type intermediate rings as recorded in [11].

######  Theorem 2.2.

For any ideal $I$ of $C(X)$, $C^{*}(X)+I=\\{f+g:f\in C^{*}(X)\text{and }g\in
I\\}$ is a $C$-type intermediate ring.

We are now ready to establish the first main result of the present paper.

######  Theorem 2.3.

Each real compact space lying between $X$ and $\beta X$ can be achieved as a
space of the form $\upsilon_{C_{\mathcal{P}}}X$ for some
$\mathcal{P}\in\Omega(X)$ if and only if $X$ is pseudocompact.

###### Proof.

If $X$ is pseudocompact then $\beta X=\upsilon X$, so that $\beta X$ is the
only real compact space lying between $X$ and $\beta X$ and we can write
$\beta X=\upsilon_{C_{\mathcal{P}}}X$, where we can take
$\mathcal{P}\equiv\mathcal{K}$. To prove the other part of this theorem assume
that $X$ is not pseudocompact. We shall construct a real compact space lying
between $X$ and $\beta X$ which is not of the form
$\upsilon_{C_{\mathcal{P}}}X$ for a $\mathcal{P}\in\Omega(X)$. As $X$ is not
pseudocompact, there exists a copy of $\mathbb{N}=\\{1,2,3,.....\\}$,
$C$-embedded in $X$. We make a partition of $\mathbb{N}$ as follows:
$\mathbb{N}=\cup_{k\in\omega_{\circ}}\mathbb{N}_{k}$, where each
$\mathbb{N}_{k}$ is an infinite set and for $k\neq j$,
$\mathbb{N}_{k}\cap\mathbb{N}_{j}=\phi$. Since $X$ is $C$-embedded in
$\upsilon X$ it follows that for each $k\in\omega_{\circ}$, $\mathbb{N}_{K}$
is $C$-embedded in $\upsilon X$. As every $C$-embedded countable subset of a
Tychonoff space $Y$ is a closed subset of $Y$ [see Ex. 3B3, Chapter 3,
Gillman-Jerison text [12]],it turns out that each $\mathbb{N}_{k}$ is a closed
subset of $\upsilon X$. Since $\mathbb{N}_{k}$ is noncompact and therefore not
closed in $\beta X$, it follows that $\operatorname{cl}_{\beta
X}\mathbb{N}_{k}\setminus\mathbb{N}_{k}\subseteq\beta X\setminus\upsilon X$.
Since disjoint zero sets in a Tychonoff space $Y$ have their closures in
$\beta Y$ disjoint, this further implies that $\\{\operatorname{cl}_{\beta
X}\mathbb{N}_{k}\setminus\mathbb{N}_{k}:k\in\omega_{\circ}\\}$, constitutes a
countably infinite family of pairwise disjoint nonempty subsets of $\beta
X\setminus\upsilon X$, indeed each such subset is a copy of
$\beta\mathbb{N}\setminus\mathbb{N}$. For each $k\in\omega_{\circ}$, we select
a point $p_{k}\in\operatorname{cl}_{\beta
X}\mathbb{N}_{k}\setminus\mathbb{N}_{k}$. Let $T=\\{p_{1},p_{2},...\\}$ and
recall that $C_{T}=\\{f\in C(X):f^{*}(p_{k})\in\mathbb{R}\text{ for each
}k\in\omega_{\circ}\\}$. It follows from Theorem 2.1 that $C_{T}$ is a
$C$-type intermediate ring. We now assert that there does not exist any ideal
$I$ of $C(X)$ such that $C_{T}=C^{*}(X)+I$. We argue by contradiction and
suppose $C_{T}=C^{*}(X)+I$ for some ideal $I$ of $C(X)$. Since $\mathbb{N}$ is
$C$-embedded in $X$, there exists an $f\in C(X)$ such that
$f(\mathbb{N}_{k})=k$ for each $k\in\omega_{\circ}$. Consequently
$f^{*}(\operatorname{cl}_{\beta X}\mathbb{N}_{k})=k$ for each
$k\in\omega_{\circ}$ in particular $f^{*}(p_{k})=k$. Thus $f^{*}$ takes real
values on the whole of $T$, hence $f\in C_{T}$. So we can write
$f=f_{1}+f_{2}$, where $f_{1}\in C^{*}(X)$ and $f_{2}\in I$. Now for a
sufficiently large value of $k$, we can write $f^{*}(p_{k})>1$. But as $p_{k}$
is a hyperreal point of $\beta X$ i.e. $p_{k}\in\beta X\setminus\upsilon X$,
we can find out an $u\in C(X)$ with $Z(u)=\phi$ such that
$u^{*}(p_{k})=\infty$. It follows that $(uf_{2})^{*}(p_{k})=\infty$. On the
other hand, the fact that $I$ is an ideal of $C(X)$ and $f_{2}\in I$ implies
that $uf_{2}\in I$. Therefore $uf_{2}\in C_{T}$ and hence
$(uf_{2})^{*}(p_{k})\in\mathbb{R}$. We arrive at a contradiction. Thus it is
settled that $C_{T}\neq C^{*}(X)+I$ for any ideal $I$ of $C(X)$. To complete
this theorem we shall prove that the realcompact space
$\upsilon_{C_{T}}X\equiv\\{p\in\beta X:f^{*}(p)\in\mathbb{R}\text{ for each
}f\in C_{T}\\}$ cannot be expressed in the form $\upsilon_{C_{\mathcal{P}}}X$
for any $\mathcal{P}\in\Omega(X)$. This time also we argue by contradiction
and assume that there is an ideal $\mathcal{P}$ of closed sets in $X$ for
which $\upsilon_{C_{T}}X=\upsilon_{C_{\mathcal{P}}}X$. It follows that
$\upsilon_{C_{T}}X=\upsilon_{C_{\mathcal{P}}(X)+C^{*}(X)}X$, this means that
$C_{T}$ and $C_{\mathcal{P}}(X)+C^{*}(X)$ belong to the same equivalence class
of $\Sigma(X)$modulo the relation ${}^{\prime}\sim^{\prime}$, introduced
earlier. But $C_{T}$ is a $C$-type intermediate ring as noted above and it
follows from the Theorem 2.2 that $C_{\mathcal{P}}(X)+C^{*}(X)$ is also a
$C$-type intermediate ring. Since each $C$-type intermediate ring happens to
be the largest member of its own equivalence class as realised in Theorem 2.1,
this implies that $C_{T}=C_{\mathcal{P}}(X)+C^{*}(X)$ a contradiction. ∎

######  Remark 2.4.

The above proof clearly suggests that a space $X$ is pseudocompact if and only
if each $C$-type ring lying between $C^{*}(X)$ and $C(X)$ is of the form
$C^{*}(X)+I$ for some ideal $I$ of $C(X)$.

##  3\. When does the class of locally-$\mathcal{P}$ spaces$/$almost
locally-$\mathcal{P}$ spaces becomes closed under product?

The following result gives a description of the members of products of ideals
of closed sets. The proof of this fact is routine.

######  Theorem 3.1.

Let $X$ be the product of an indexed family
$\\{X_{\alpha}:\alpha\in\Lambda\\}$ of topological spaces and
$\mathcal{P}\equiv\prod_{\alpha\in\Lambda}\mathcal{P}_{\alpha}$, where for
each $\alpha\in\Lambda,~{}\mathcal{P}_{\alpha}\in\Omega(X_{\alpha})$. Then
$\mathcal{P}=\\{A\subseteq X:A\text{ is closed in }X\text{ and
}A\subseteq\prod_{\alpha\in\Lambda}A_{\alpha}$, where
$A_{\alpha}\in\mathcal{P}_{\alpha}\text{ for each }\alpha\in\Lambda\\}$.

With the notation of Theorem 3.1, the next proposition tells us precisely when
the product of locally-$\mathcal{P}_{\alpha}$-spaces becomes
locally-$\mathcal{P}$.

######  Theorem 3.2.

$X=\prod_{\alpha\in\Lambda}X_{\alpha}$ is a locally-$\mathcal{P}$space if and
only if

  1. (1)

$X_{\alpha}$ is locally-$\mathcal{P}_{\alpha}$ for each $\alpha\in\Lambda$.

  2. (2)

$X_{\alpha}\in\mathcal{P}_{\alpha}$ for all but possibly finitely many
$\alpha$’s in $\Lambda$.

###### Proof.

Let $X$ be locally-$\mathcal{P}$, choose a point $x_{\beta}$ from the space
$X_{\beta},\beta\in\Lambda$. Let $x$ be a point in $X$, whose $\beta$ th co-
ordinate is $x_{\beta}$. On using Theorem 3.1, we can find out an open
neighbourhood $U$ of $x$ in $X$ such that
$\operatorname{cl}_{X}U\subseteq\prod_{\alpha\in\Lambda}A_{\alpha}$, where
$A_{\alpha}\in\mathcal{P}_{\alpha}$ for each $\alpha\in\Lambda$. We can write
$U=\prod_{\alpha\in\Lambda}U_{\alpha}$, where each $U_{\alpha}$ is open in
$X_{\alpha}$ and $U_{\alpha}\neq X_{\alpha}$ for at most finitely many
$\alpha$’s in $\Lambda$. It follows that
$\prod_{\alpha\in\Lambda}\operatorname{cl}_{X_{\alpha}}U_{\alpha}\subseteq\prod_{\alpha\in\Lambda}A_{\alpha}$
and hence $\operatorname{cl}_{X_{\beta}}U_{\beta}\subseteq A_{\beta}$ implying
that $\operatorname{cl}_{X_{\beta}}U_{\beta}\in\mathcal{P}_{\beta}$. Thus
$X_{\beta}$ becomes locally-$\mathcal{P}_{\beta}$ and it is easy to see that
$A_{\alpha}=X_{\alpha}$ for all but finitely many $\alpha$’s and consequently
$X_{\alpha}\in\mathcal{P}_{\alpha}$ for all these $\alpha$’s.

Conversely, let $X_{\alpha}$ be locally-$\mathcal{P}_{\alpha}$ for each
$\alpha\in\Lambda$ and let there be a finite subset
$\Lambda_{\circ}=\\{\alpha_{1},\alpha_{2},...,\alpha_{n}\\}$ of $\Lambda$ for
which $X_{\alpha}\in\mathcal{P}_{\alpha}$ for all
$\alpha\in\Lambda\setminus\Lambda_{\circ}$. Choose a point $x$ from $X$, with
its co-ordinate $x_{\beta}$ for $\beta\in\Lambda$. For each
$\alpha\in\Lambda\setminus\Lambda_{\circ}$, we select an open neighbourhood
$V_{\alpha}$ of $x_{\alpha}$ in $X_{\alpha}$ such that
$\operatorname{cl}_{X_{\alpha}}V_{\alpha}\in\mathcal{P}_{\alpha}$. Let
$V=\prod_{\alpha\in\Lambda}V_{\alpha}$, with $V_{\alpha}=X_{\alpha}$ whenever
$\alpha\in\Lambda\setminus\Lambda_{\circ}$. Then $V$ is an open neighbourhood
of $x$ in $X$ and
$\operatorname{cl}_{X}V=\prod_{\alpha\in\Lambda}\operatorname{cl}_{X_{\alpha}}V_{\alpha}\in\mathcal{P}$.
Thus $X$ becomes locally-$\mathcal{P}$. ∎

The following fact follows by adopting similar arguments:

######  Theorem 3.3.

$X=\prod_{\alpha\in\Lambda}X_{\alpha}$ is an almost
locally-$\mathcal{P}$-space if and only if

  1. (1)

$X_{\alpha}$ is almost locally-$\mathcal{P}_{\alpha}$ for each
$\alpha\in\Lambda$ and

  2. (2)

$X_{\alpha}\in\mathcal{P}_{\alpha}$ for all but finitely many $\alpha$’s in
$\Lambda$.

We shall now record a particularly interesting special case of Theorem 3.2 and
3.3 For this purpose we need to recall that a subset $A$ of $X$ is called
bounded if each $f\in C(X)$ is bounded on $A$. Every pseudocompact subset of
$X$ is bounded. The following theorem of Mandelkar, proved in 1971 inform us
that the converse of the last statement is true for a special class of subsets
of $X$.

######  Theorem 3.4.

([14]) A support in $X$ i.e. a subset of the form
$\operatorname{cl}_{X}(X-Z(f)),f\in C(X)$ is bounded if and only if it is
pseudocompact.

We call a space $X$ locally pseudocompact( almost locally pseudocompact) if
each point $x$ on it has an open neighbourhood $U_{x}$ whose closure is
pseudocompact(respectively each nonempty open set $U$ in $X$ contains a
nonempty open set $V$ such that $\operatorname{cl}_{X}V$ is pseudocompact).

Let us call a space $X$ locally bounded ( almost locally bounded ) if $X$ is
locally-$\mathcal{B}$ ( respectively almost locally-$\mathcal{B}$ ), where
$\mathcal{B}$ is the ideal of all closed bounded sets in $X$. Since each
neighbourhood of a point in $X$ contains a cozero set neighbourhood of the
same point, the following proposition is an easy consequence of Theorem 3.4.

######  Theorem 3.5.

$X$ is locally pseudocompact/ almost locally pseudocompact if and only if $X$
is locally bounded ( respectively almost locally bounded).

The following propositions now emerge as special cases of Theorem 3.2 and 3.3
respectively.

######  Theorem 3.6.

The product space $X\equiv\prod_{\alpha\in\Lambda}X_{\alpha}$ is locally
pseudocompact if and only if each $X_{\alpha}$ is locally pseudocompact and
for all but finitely many $\alpha$’s, $X_{\alpha}$’s are pseudocompact.

######  Theorem 3.7.

The product space $X\equiv\prod_{\alpha\in\Lambda}X_{\alpha}$ is almost
locally pseudocompact if and only if each $X_{\alpha}$ is almost locally
pseudocompact and for all but finitely many $\alpha$’s, $X_{\alpha}$’s are
pseudocompact.

We would like to mention at this point that, since the closed pseudocompact
subspaces of a space $X$ may not make an ideal of closed sets, indeed a closed
subset of a pseudocompact space need not be pseudocompact (vide: the right
edge $\\{\omega_{1}\\}\times\omega_{\circ}$ of the Tychonoff plank
$T\equiv[0,\omega_{1}]\times[0,\omega_{\circ}]\setminus\\{(\omega_{1},\omega_{\circ})\\}$
is not pseudocompact), we cannot deduce the Theorem 3.5 and 3.6 directly from
the general propositions 3.2 and 3.3 respectively. Incidentally the following
example of an almost locally compact space which is not locally compact is
constructed in [4].

######  Example 3.8.

Let $X$ be an uncountable set, each point of which excepting a point $p\in X$
is isolated and a typical neighbourhood of $p$ is a set $A$ containing $p$
such that $X\setminus A$ is at most countable set.

The following example is quite analogous to the above one.

######  Example 3.9.

Let $X$ be an uncountable set with $\lvert X\rvert>\omega_{1}$ with the
following topology: each point of $X$, barring a distinguished point $p$ is
isolated and a typical neighbourhood of $p$ is a set $A$ containing $p$ for
which $\lvert X\setminus A\rvert\leq\omega_{1}$. It is easy to check that if
$\mathcal{P}$ is any one of the three ideals of closed sets viz
$\mathfrak{F},\mathfrak{B}$ and $\mathfrak{L}$, then $X$ is almost
locally-$\mathcal{P}$ without being locally-$\mathcal{P}$. Here $\mathfrak{F}$
stands for the ideal of all finite subsets of $X$ and $\mathfrak{L}$ that for
all closed Lindelof subsets of $X$.

##  4\. On the question of freeness and essentiality of ideals ideals
$C_{\mathcal{P}}(X)$ and $C^{\mathcal{P}}_{\infty}(X)$

The following proposition relates the local $\mathcal{P}$-condition on $X$
with the freeness of the ideals $C_{\mathcal{P}}(X)$ and
$C^{\mathcal{P}}_{\infty}(X)$.

######  Theorem 4.1.

For $\mathcal{P}\in\Omega(X)$, the following three statements are equivalent:

  1. (1)

$X$ is locally-$\mathcal{P}$.

  2. (2)

$C_{\mathcal{P}}(X)$ is a free ideal of $C(X)$.

  3. (3)

$C^{\mathcal{P}}_{\infty}(X)$ is a free ideal of
$C^{\mathcal{P}}_{\infty}(X)+C^{*}(X)$.

###### Proof.

$(1)\Rightarrow(2):$ Let $X$ be locally-$\mathcal{P}$ and $x\in X$. Then there
exists an open neighbourhood $U$ of $x$ such that
$\operatorname{cl}_{X}U\in\mathcal{P}$. By using the complete regularity of
$X$, we can find an $f\in C^{*}(X)$ such that $f(X\setminus U)=0$ and
$f(x)=1$. This implies that $X\setminus Z(f)\subseteq U$ and consequently
$\operatorname{cl}_{X}(X\setminus Z(f))\subseteq\operatorname{cl}_{X}U$ and
hence $\operatorname{cl}_{X}(X\setminus Z(f))\in\mathcal{P}$ i.e. $f\in
C_{\mathcal{P}}(X)$, we note that $f(x)\neq 0$.

$(2)\Rightarrow(3):$ is trivial as $C_{\mathcal{P}}(X)\subseteq
C^{\mathcal{P}}_{\infty}(X)$.

$(3)\Rightarrow(1):$ Let $(3)$ hold and $x\in X$. Then there exists $f\in
C^{\mathcal{P}}_{\infty}(X)$ such that $f(x)\neq 0$, say $\lvert
f(x)\rvert=\lambda$. Take $U=\\{y\in X:\lvert
f(y)\rvert>\frac{\lambda}{2}\\}$, which is a nonempty open set. Since
$\operatorname{cl}_{X}U\subseteq\\{y\in X:\lvert
f(y)|\geq\frac{\lambda}{2}\\}$ and the last set is a member of $\mathcal{P}$,
it follows that $\operatorname{cl}_{X}U\in\mathcal{P}$. Thus $X$ becomes
locally-$\mathcal{P}$. ∎

We would like to mention that a variant condition using a different
terminology both necessary and sufficient for the freeness of the ideal
$C_{\mathcal{P}}(X)$ in $C(X)$ have also been discovered in [[20], proposition
2.11].

######  Remark 4.2.

With $\mathcal{P}\equiv\mathcal{K}\equiv$ the ideal of compact sets in $X$,
the above theorem reads: $X$ is locally compact if and only if $C_{K}(X)$ is a
free ideal of $C(X)$ if and only if $C_{\infty}(X)$ is a free ideal of
$C^{*}(X)$, a standard result in Rings of Continuous Functions [see [12]].

######  Remark 4.3.

On choosing $\mathcal{P}\equiv\mathcal{B}$, it follows from Mandelkar’s
theorem 3.4 that $C_{\psi}(X)=C_{\mathcal{B}}(X)$ and
$C^{\psi}_{\infty}(X)=C^{\mathcal{B}}_{\infty}(X)$. Thus with this particular
choice of $\mathcal{P}$, Theorem 4.1 states that $X$ is locally pseudocompact
when and only when $C_{\psi}(X)$ is a free ideal of $C(X)$, equivalently
$C^{\psi}_{\infty}(X)$ is a free ideal of $C^{*}(X)$.

Before embarking on the question of essentiality of the ideals
$C_{\mathcal{P}}(X)$ and $C^{\mathcal{P}}_{\infty}(X)$, we write down the
following topological description of essential ideals of an intermediate ring.
The proof is accomplished by following the technique of [4].

######  Theorem 4.4.

A nonzero ideal $I$ in an intermediate ring $A(X)$ is essential in this ring
if and only if $\cap Z[I]\equiv\cap_{f\in I}Z(f)$ is a nowhere dense subset of
$X$.

It follows from this Theorem that every free ideal of $A(X)$ is an essential
ideal. Also for any $\mathcal{P}\in\Omega(X)$, the local $\mathcal{P}$
condition on $X$ is stronger than almost local $\mathcal{P}$. The following
proposition which correlates almost local $\mathcal{P}$ requirement with
essentiality of the ideals $C_{\mathcal{P}}(X)$ and
$C^{\mathcal{P}}_{\infty}(X)$ seems therefore natural.

See also the proposition 3.2 in [20] in this context.

######  Theorem 4.5.

For any $\mathcal{P}\in\Omega(X)$, the following statements are equivalent:

  1. (1)

$X$ is almost locally-$\mathcal{P}$.

  2. (2)

$C_{\mathcal{P}}(X)$ is an essential ideal of $C(X)$.

  3. (3)

$C^{\mathcal{P}}_{\infty}(X)$ is an essential ideal of
$C^{*}(X)+C^{\mathcal{P}}_{\infty}(X)$.

###### Proof.

$(1)\Rightarrow(2):$ Suppose that $X$ is almost locally-$\mathcal{P}$. Let
$f\neq 0$ be a non unit function in $C(X)$. We shall show that there is a
$g\in C_{\mathcal{P}}(X)$ such that $fg\neq 0$. By using the regularity of $X$
and the local $\mathcal{P}$ condition on it, we can find nonempty open sets
$A$ and $B$ in $X$ such that $\operatorname{cl}_{X}B\subseteq
A\subseteq\operatorname{cl}_{X}A\subseteq X\setminus Z(f)$ and
$\operatorname{cl}_{X}B\in\mathcal{P}$. We now use the complete regularity of
$X$ to produce a $g\in C^{*}(X)$ such that $g(X\setminus B)=0$ and $g(b)=1$
for some point $b\in B$. Hence $X\setminus Z(g)\subseteq B$. Consequently
$\operatorname{cl}_{X}(X\setminus Z(g))\subseteq\operatorname{cl}_{X}B$. Since
$\operatorname{cl}_{X}B\in\mathcal{P}$, it follows that
$\operatorname{cl}_{X}(X\setminus Z(g))\mathcal{P}$ i.e. $g\in
C_{\mathcal{P}}(X)$. We observe that $f(b)g(b)\neq 0$.

$(2)\Rightarrow(3):$ This is trivial because $C_{\mathcal{P}}(X)\subseteq
C^{\mathcal{P}}_{\infty}(X)$.

$(3)\Rightarrow(1):$ The proof is accomplished on using Theorem 4.4, taking
care of the fact that if $f\in C^{\mathcal{P}}_{\infty}(X)$ and $g\in
C^{*}(X)$, then $fg\in C^{\mathcal{P}}_{\infty}(X)$ and following the
arguments in [5], closely. ∎

The following proposition is a consequence of Theorem 3.5, first two lines in
Remark 4.3 and Theorem 4.5.

######  Theorem 4.6.

The statements written below are equivalent.

  1. (1)

$X$ is almost locally pseudocompact.

  2. (2)

$C_{\psi}(X)$ is an essential ideal of $C(X)$.

  3. (3)

$C^{\psi}_{\infty}(X)$ is an essential ideal of $C^{*}(X)$.

The authors in [13], have observed that for a large class of spaces $X$, the
socle $S$ of $C(X)$ turns out to be the same as the ring $C_{K}(X)$ and have
asked for a characterisation of space $X$ with such property. Motivated by
this question, the authors in [4] offered a simple topological
characterization of such spaces and a special subclass of those spaces in the
following manner.

######  Theorem 4.7.

(Theorem 4.5[4])

  1. (1)

$C_{K}(X)=S$ if and only if $X$ is pseudodiscrete meaning that each compact
subset of $X$ has finite interior.

  2. (2)

$C_{\infty}(X)=S$ if and only if $X$ is pseudodiscrete and contains at most a
finite numbers of isolated points.

Being prompted by this result, we have established the following fact which
places this result in a more general setting. Given $\mathcal{P}\in\Omega(X)$,
we call $X$, $\mathcal{P}$-pseudodiscrete if each member of $\mathcal{P}$ has
finite interior.

######  Theorem 4.8.

Let $\mathcal{P}\in\Omega(X)$. Suppose that either $X$ has no isolated point
or each singleton subset of $X$ is a member of $\mathcal{P}$. Then

  1. (1)

$C_{\mathcal{P}}(X)=S$ if and only if $X$ is $\mathcal{P}$ pseudodiscrete.

  2. (2)

$C^{\mathcal{P}}_{\infty}(X)=S$ if and only if $X$ is $\mathcal{P}$
pseudodiscrete and contains at most a finite numbers of isolated points.

###### Proof.

  1. (1)

Since the socle $S$ of $C(X)$ contains precisely those functions in $C(X)$,
which vanish everywhere except on a finite number of points of $X$, as is
established in [13], Proposition 3.3, it follows on using the hypothesis at
the beginning of the statement of this theorem that $S\subseteq
C_{\mathcal{P}}(X)$. The reverse containment $C_{\mathcal{P}}\subseteq S$ can
be proved by using the $\mathcal{P}$-pseudodiscreteness of $X$ and by adapting
carefully the arguments in the proof of Theorem 4.7 (1).

  2. (2)

This can be proved by closely following the arguments of proof of Theorem 4.7
(2) and making certain obvious modifications.

∎

We would like to record a special case of Theorem 4.8, on using
$\mathcal{P}\equiv\mathcal{B}$. To do so, we need the following fact.

######  Theorem 4.9.

The pair of statements written below are equivalent:

  1. (1)

Every closed pseudocompact subset of $X$ has finite interior.

  2. (2)

Every closed bounded subset of $X$ has finite interior.

###### Proof.

$(2)\Rightarrow(1):$ This is trivial. Assume that $(1)$ holds. Take a closed
bounded set $A$ in $X$. If possible let $int_{X}A$ be infinite. We choose a
countable infinite subset $B=\\{b_{1},b_{2},...,b_{n},....\\}$ of $int_{X}A$.
For each $i\in\mathbb{N}$, there exists an $f_{i}\in C(X)$ such that $b_{i}\in
X\setminus Z(f_{i})\subseteq int_{X}A$. This implies that
$B\subseteq\cup_{i=1}^{\infty}(X\setminus Z(f_{i}))\subseteq int_{X}A$, hence
$B\subseteq X\setminus\cap_{i=1}^{\infty}Z(f_{i})=X\setminus Z(f)\subseteq
int_{X}A\subseteq A$ for some $f\in C(X)$. This implies that $B\subseteq
X\setminus Z(f)\subseteq\operatorname{cl}_{X}(X\setminus Z(f))\subseteq A$.
Now since $A$ is bounded so also is $\operatorname{cl}_{X}(X\setminus Z(f))$,
which in view of Theorem 3.4, turns out to be a closed pseudocompact space
with infinite interior. A contradiction is arrived at. ∎

######  Theorem 4.10.

For a space $X$,

  1. (1)

$C_{\psi}(X)=S$ if and only if every closed pseudocompact subset of $S$ has
finite interior.

  2. (2)

$C^{\psi}_{\infty}(X)=S$ if and only if every closed pseudocompact subset of
$X$ has finite interior and $X$ contains not more than finitely many isolated
points.

###### Proof.

Follows from Theorem 4.8 and Theorem 4.9. ∎

######  Example 4.11.

The conclusions of Theorem 4.8 may not hold good if the hypothesis that each
singleton subset of $X$ is a member of $\mathcal{P}$(in case $X$ contains at
least one isolated point) is dropped. Indeed we take any infinite Tychonoff
space $X$, where there is just one isolated point say $p$. Let $\mathcal{P}$
be any member of $\Omega(X)$ with the following two conditions:

  1. (1)

If $A\in\mathcal{P}$ then $p\notin A$.

  2. (2)

Interior of each member of $\mathcal{P}$ is finite.

To be specific we can choose $\mathcal{P}$ to be the family of all finite
subsets of $X$ which miss the point $p$ or the family of all closed nowhere
dense subsets of $X$, missing the point $p$. The function $f\in C(X)$ given by
$f(p)=1$ and $f(q)=0$ for all $p\neq q$ in $X$, belongs to the socle $S$ of
$C(X)$ in view of Proposition 3.3 in [13]. But we see that
$\operatorname{cl}_{X}(X\setminus Z(f))=\\{p\\}$, which does not belong to
$\mathcal{P}$, which means that $f\notin C_{\mathcal{P}}(X)$. Altogether
$S\neq C_{\mathcal{P}}(X)$. It is also easy to check that $f\notin
C^{\mathcal{P}}_{\infty}(X)$ because $\\{x\in X:\lvert
f(x)\rvert\geq\frac{1}{2}\\}=\\{p\\}$ does not belong to $\mathcal{P}$. Then
$S\neq C^{\mathcal{P}}_{\infty}(X)$.

##  5\. Ideals $C_{\mathcal{P}}(X)$ are the same as $z_{\circ}$-ideals in
$C(X)$

For an element $a$ of a commutative ring $R$ with identity, let $P_{a}$ stand
for the intersection of all minimal prime ideals of $R$ which contain $a$. An
ideal $I$ of $R$ is called a $z_{\circ}$-ideal if for each member $a$ of
$I,P_{a}\subseteq I$.

The following fact is proved in [6].

######  Theorem 5.1.

For $f\in C(X),P_{f}=\\{g\in C(X):int_{X}Z(f)\subseteq int_{X}Z(g)\\}$.

We now prove the main result of this section.

######  Theorem 5.2.

A proper ideal $I$ of $C(X)$ is a $z^{\circ}$-ideal if and only if there
exists $\mathcal{P}\in\Omega(X)$ such that $I=C_{\mathcal{P}}(X)$.

###### Proof.

Let $I$ be a $z^{\circ}$-ideal of $C(X)$. Suppose $\mathcal{P}[I]$ is the
smallest member of $\Omega(X)$ containing the family of sets
$\\{\operatorname{cl}_{X}(X\setminus Z(f)):f\in I\\}$. Since the family is
already closed under finite intersection, as can be easily verified, it
follows that $\mathcal{P}[I]=\\{Y\subseteq X:Y\text{ is closed in }X\text{ and
}Y\subseteq\operatorname{cl}_{X}(X\setminus Z(f))\text{ for some }f\in I\\}$.
We shall show that $I=C_{\mathcal{P}[I]}(X)$. It is plain that $I\subseteq
C_{\mathcal{P}[I]}(X)$. To prove the other containment, choose $f$ from
$C_{\mathcal{P}[I]}(X)$. Then there exists a $g\in I$ such that
$\operatorname{cl}_{X}(X\setminus
Z(f))\subseteq\operatorname{cl}_{X}(X\setminus Z(g))$ which clearly implies
that $int_{X}Z(g)\subseteq int_{X}Z(f)$. It follows from the Theorem 5.1 that
$f\in P_{g}$. Also the fact that $I$ is a $z^{\circ}$-ideal of $C(X)$ together
with $g\in I$ imply that $P_{g}\subseteq I$. Hence $f\in I$. Thus
$C_{\mathcal{P}[I]}(X)\subseteq I$.

To prove the other part of this theorem, let $\mathcal{P}\in\Omega(X)$. We
shall show that $C_{\mathcal{P}}(X)$ is a $z^{\circ}$-ideal of $C(X)$. Choose
$f\in C_{\mathcal{P}}(X)$ and $g\in P_{f}$. Then from Theorem 5.1, we can
write $int_{X}Z(f)\subseteq int_{X}Z(g)$. This implies that
$\operatorname{cl}_{X}(X\setminus
Z(g))\subseteq\operatorname{cl}_{X}(X\setminus Z(f))$. But $f\in
C_{\mathcal{P}}(X)$ means that $\operatorname{cl}_{X}(X\setminus
Z(f))\in\mathcal{P}$. Since $\mathcal{P}$ is an ideal of closed sets in $X$,
it follows that $\operatorname{cl}_{X}(X\setminus Z(g))\in\mathcal{P}$ and
hence $g\in C_{\mathcal{P}}(X)$. Thus we get $P_{f}\subseteq
C_{\mathcal{P}}(X)$ for an arbitrary $f$ in $C_{\mathcal{P}}(X)$. This settles
that $C_{\mathcal{P}}(X)$ is a $z^{\circ}$-ideal of $C(X)$. ∎

######  Corollary 5.3.

Let an ideal $I$ in $C(X)$ contain divisors of zero only. Then $I$ is
contained in a proper $z^{\circ}$-ideal of $C(X)$.

###### Proof.

It follows from the proof of Theorem 5.2, first part that $I\subseteq
C_{\mathcal{P}[I]}(X)$. Since each $f\neq 0$ in $I$ is a divisor of zero, it
follows that $int_{X}Z(f)$ is nonempty proper open subset of $X$. Consequently
for any such $f,\operatorname{cl}_{X}(X\setminus Z(f))$ is a nonempty proper
closed subset of $X$ and therefore $X\notin\mathcal{P}[I]$. It follows that
$1\notin C_{\mathcal{P}[I]}(X)$. ∎

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