* $\backslash$AlCapFnt item $\backslash$SetAlgoCaptionLayout{style}, item $\backslash$ProcFnt{$<$text$>$}, item $\backslash$AlCapSty{$<$text$>$}, item $\backslash$AlCapSty{$<$text$>$}, item $\backslash$AlCapFnt{$<$text$>$}, item $\backslash$AlCapFnt{$<$text$>$}, item $\backslash$AlCapFnt{$<$text$>$}, item $\backslash$AlCapFnt{$<$text$>$}, §9.5.2, §9.5.2, §9.5.2, §9.5.2, §9.5.2, §9.5.3
  * $\backslash$AlCapNameFnt item $\backslash$SetAlgoCaptionLayout{style}, item $\backslash$ProcArgFnt{$<$text$>$}, item $\backslash$AlCapNameSty{$<$text$>$}, item $\backslash$AlCapNameSty{$<$text$>$}, item $\backslash$AlCapNameFnt{$<$text$>$}, item $\backslash$AlCapNameFnt{$<$text$>$}, item $\backslash$AlCapNameFnt{$<$text$>$}, item $\backslash$AlCapNameFnt{$<$text$>$}, §9.5.2, §9.5.2, §9.5.2, §9.5.2
  * $\backslash$AlCapNameSty item $\backslash$SetProcArgSty{$<$font$>$}, item $\backslash$SetAlCapNameSty{$<$commandname$>$}:, item $\backslash$ProcArgSty{$<$text$>$}, item $\backslash$AlCapNameSty{$<$text$>$}, item $\backslash$AlCapNameSty{$<$text$>$}, item $\backslash$AlCapNameSty{$<$text$>$}, item $\backslash$AlCapNameSty{$<$text$>$}, item $\backslash$AlCapNameFnt{$<$text$>$}, item $\backslash$AlCapNameFnt{$<$text$>$}, item $\backslash$ProcNameSty{$<$text$>$}, item $\backslash$ProcArgSty{$<$text$>$}, §9.5.2, §9.5.2, §9.5.2, §9.5.2
  * $\backslash$AlCapSkip item $\backslash$AlCapSkip, item $\backslash$AlCapSkip, item $\backslash$SetAlCapSkip{length}
  * $\backslash$AlCapSty item $\backslash$SetProcNameSty{$<$font$>$}, item $\backslash$SetAlCapSty{$<$commandname$>$}:, item $\backslash$ProcNameSty{$<$text$>$}, item $\backslash$AlCapSty{$<$text$>$}, item $\backslash$AlCapSty{$<$text$>$}, item $\backslash$AlCapSty{$<$text$>$}, item $\backslash$AlCapSty{$<$text$>$}, item $\backslash$AlCapFnt{$<$text$>$}, item $\backslash$AlCapFnt{$<$text$>$}, item $\backslash$ProcSty{$<$text$>$}, §9.5.2, §9.5.2, §9.5.2, §9.5.2, §9.5.2
  * $\backslash$AlFnt item $\backslash$AlFnt, item $\backslash$AlFnt
  * $\backslash$AlgoDisplayBlockMarkers item displayblockmarkers, item displayblockmarkers, item 7, item $\backslash$AlgoDisplayBlockMarkers, item $\backslash$AlgoDisplayBlockMarkers, item $\backslash$AlgoDisplayGroupMarkers
  * $\backslash$AlgoDisplayGroupMarkers item displaygroupmarkers, item displaygroupmarkers, item $\backslash$AlgoDisplayGroupMarkers, item $\backslash$AlgoDisplayGroupMarkers
  * $\backslash$AlgoDisplaygroupMarkers item 7
  * $\backslash$algomargin item $\backslash$algomargin, item $\backslash$algomargin, item $\backslash$algomargin, item $\backslash$IncMargin{length}, item $\backslash$DecMargin{length}, item $\backslash$DecMargin{length}
  * algomathdisplay §9.4
  * algonl item algonl, item algonl
  * algoruled item algoruled:, item algoruled:, item $\backslash$RestyleAlgo{style}, item $\backslash$NoCaptionOfAlgo
  * $\backslash$AlTitleFnt item $\backslash$AlTitleSty{$<$text$>$}, item $\backslash$AlTitleSty{$<$text$>$}, item $\backslash$AlTitleFnt{$<$text$>$}, item $\backslash$AlTitleFnt{$<$text$>$}, item $\backslash$AlTitleFnt{$<$text$>$}
  * $\backslash$AlTitleSty item $\backslash$SetAlTitleSty{$<$commandname$>$}, item $\backslash$AlTitleSty{$<$text$>$}, item $\backslash$AlTitleSty{$<$text$>$}, item $\backslash$AlTitleSty{$<$text$>$}, item $\backslash$AlTitleFnt{$<$text$>$}, item $\backslash$AlTitleFnt{$<$text$>$}
  * $\backslash$Altrimenti 5th item
  * $\backslash$AltSe 5th item
  * $\backslash$Anderes 6th item
  * arabtex §5
  * $\backslash$ArgSty item $\backslash$SetKwArray{Kw}{array}, item $\backslash$SetKwFunction{KwFn}{Fn}, item $\backslash$ArgSty{$<$text$>$}, item $\backslash$ArgSty{$<$text$>$}
  * article-hermes item article-hermes, item article-hermes
  * ascelike 1st item
  * $\backslash$até 2nd item
  * $\backslash$Aus 1st item
  * $\backslash$autoref item $\backslash$SetAlgorithmName{algorithmname}{algorithmautorefname}{list of algorithms name}, item $\backslash$SetAlgoProcName{aname}{anautorefname}, item $\backslash$SetAlgoFuncName{aname}{anautorefname}
  * $\backslash$Autre item 5
  * $\backslash$Begin 1st item, 2nd item, item $\backslash$SetKwBlock{Begin}{begin}{end}, item $\backslash$SetKwBlock{Begin}{begin}{end}
  * $\backslash$Beginn 3rd item
  * $\backslash$Bis 2nd item
  * $\backslash$BlankLine item $\backslash$BlankLine, item $\backslash$BlankLine
  * $\backslash$BlockMarkersSty item 7, item $\backslash$BlockMarkersSty{$<$text$>$}, item $\backslash$BlockMarkersSty{$<$text$>$}
  * boxed item boxed:, item boxed:, item $\backslash$RestyleAlgo{style}
  * boxruled item boxruled:, item boxruled:
  * $\backslash$caption 5th item, 5th item, item $\backslash$caption{thetitle}, item $\backslash$caption{thetitle}, item $\backslash$TitleOfAlgo{thetitle}, item $\backslash$ProcArgFnt{$<$text$>$}, item $\backslash$AlCapNameSty{$<$text$>$}, item $\backslash$AlCapNameFnt{$<$text$>$}, item $\backslash$ProcNameSty{$<$text$>$}, item $\backslash$ProcArgSty{$<$text$>$}, item $\backslash$ProcNameFnt{$<$text$>$}, §9.2.1, §9.5.2, §9.5.2
  * $\backslash$Cas item 5
  * $\backslash$Case 3rd item, 4th item, 5th item, 7th item, 6th item
  * $\backslash$Case(comment) 6th item, 8th item
  * $\backslash$Caso item 5, 5th item
  * $\backslash$Comment item $\backslash$SetKwComment{Comment}{start}{end}
  * $\backslash$Comment* item $\backslash$SetKwComment{Comment}{start}{end}, §11.5
  * commentsnumbered item commentsnumbered:, item commentsnumbered:, §9.3.1, §9.7
  * $\backslash$CommentSty item $\backslash$CommentSty{$<$text$>$}, item $\backslash$CommentSty{$<$text$>$}
  * croatian item croatian:, item croatian:
  * croatiankw item croatiankw:, item croatiankw:
  * czech item czech:, item czech:, item czech:, item czech:
  * czechkw item czechkw:, item czechkw:, item czechkw:, item czechkw:
  * $\backslash$DataSty item $\backslash$SetKwArray{Kw}{array}, item $\backslash$DataSty{$<$text$>$}, item $\backslash$DataSty{$<$text$>$}
  * $\backslash$Daten 1st item
  * $\backslash$Datos item 1
  * $\backslash$Deb item 3
  * $\backslash$DecMargin 8th item, item $\backslash$DecMargin{length}, item $\backslash$DecMargin{length}, item $\backslash$DecMargin{length}, item $\backslash$DecMargin{length}
  * $\backslash$decmargin 8th item
  * $\backslash$Devolver item 2
  * $\backslash$DisplayBlockMarkers item $\backslash$DisplayBlockMarkers, item $\backslash$DisplayBlockMarkers
  * displayblockmarkers item displayblockmarkers, item displayblockmarkers, item 7, item $\backslash$DisplayBlockMarkers
  * $\backslash$DisplayGroupMarkers item $\backslash$DisplayGroupMarkers, item $\backslash$DisplayGroupMarkers
  * displaygroupmarkers item displaygroupmarkers, item displaygroupmarkers, item 7, item $\backslash$DisplayGroupMarkers
  * $\backslash$Donnees item 1
  * $\backslash$DontPrintSemicolon 5th item, item $\backslash$;, item $\backslash$DontPrintSemicolon, item $\backslash$DontPrintSemicolon
  * $\backslash$dontprintsemicolon 5th item
  * $\backslash$DoThat item $\backslash$SetKwFunction{KwFn}{Fn}, item $\backslash$SetKwFunction{KwFn}{Fn}
  * dotocloa item dotocloa, item dotocloa
  * $\backslash$eIf 19th item, 20th item, 21st item, 22nd item, 23rd item, 24th item, 21st item, 27th item
  * $\backslash$Ein 1st item
  * elsart §5
  * $\backslash$Else 13rd item, 14th item, 11st item, 27th item, 9th item
  * $\backslash$Else(comment) 10th item
  * $\backslash$ElseIf 7th item, 8th item, 5th item, 7th item
  * $\backslash$ElseIf(comment) 6th item, 8th item
  * endfloat item endfloat:, item endfloat:
  * english item english:, item english:
  * $\backslash$EnOtroCaso item 4
  * $\backslash$EnOtroCasoSi item 4
  * $\backslash$Enqto 10th item
  * $\backslash$Entrada item 1
  * $\backslash$Entree item 1
  * $\backslash$Ergebnis 1st item
  * $\backslash$eSe 4th item
  * $\backslash$eSea 5th item
  * $\backslash$eSi item 4
  * $\backslash$eSSi item 4
  * $\backslash$eWenn 5th item
  * $\backslash$Fall 6th item
  * figure item figure:, item figure:
  * fillcomment item fillcomment (default), item fillcomment (default), item $\backslash$SetFillComment
  * $\backslash$Finche 11st item
  * $\backslash$Fn item $\backslash$SetKwFunction{KwFn}{Fn}
  * $\backslash$For 1st item, 2nd item, 1st item, 3rd item
  * $\backslash$For(comment) 2nd item, 4th item
  * $\backslash$ForAll 1st item, 2nd item
  * $\backslash$ForEach 1st item, 2nd item
  * algo2e §5, item french:, item french:, item algo2e:, item algo2e:, item slide:, item slide:, item algochapter:, item algochapter:, item algosection:, item algosection:, item algopart:, item algopart:, item $\backslash$TitleOfAlgo{thetitle}
  * frenchkw item frenchkw:, item frenchkw:
  * $\backslash$Fuer 7th item
  * $\backslash$FuerAlle 10th item
  * $\backslash$FuerJedes 9th item
  * $\backslash$FuerPar 8th item
  * $\backslash$FuncArgSty item $\backslash$FuncArgSty{$<$text$>$}, item $\backslash$FuncArgSty{$<$text$>$}
  * $\backslash$FuncSty item $\backslash$SetKwFunction{KwFn}{Fn}, item $\backslash$FuncSty{$<$text$>$}, item $\backslash$FuncSty{$<$text$>$}
  * german item german:, item german:
  * germankw item germankw:, item germankw:
  * $\backslash$gIf 23rd item
  * $\backslash$gSonstWenn 5th item
  * hangingcomment item hangingcomment, item hangingcomment
  * hermes §5
  * hyperref item hyperref, item hyperref, item $\backslash$SetAlgorithmName{algorithmname}{algorithmautorefname}{list of algorithms name}, item $\backslash$SetAlgoProcName{aname}{anautorefname}, item $\backslash$SetAlgoFuncName{aname}{anautorefname}
  * $\backslash$If 1st item, 2nd item, 1st item, 3rd item
  * $\backslash$If(comment) 2nd item
  * $\backslash$If(comment1) 4th item
  * $\backslash$IncMargin 7th item, item $\backslash$IncMargin{length}, item $\backslash$IncMargin{length}
  * $\backslash$incmargin 7th item
  * $\backslash$Indm item $\backslash$Indm, item $\backslash$Indm
  * $\backslash$Indp item $\backslash$Indp, item $\backslash$Indp
  * $\backslash$Inicio item 3
  * $\backslash$Iniciob 3rd item
  * $\backslash$Inizio 3rd item
  * inoutnumbered item inoutnumbered:, item inoutnumbered:
  * $\backslash$Input item $\backslash$SetKwInput{Kw}{input}
  * $\backslash$interspacetitleboxruled §9.2.2
  * $\backslash$interspacetitleruled §9.2.2
  * italiano item italiano:, item italiano:
  * italianokw item italianokw:, item italianokw:
  * $\backslash$Kw item $\backslash$SetKwInput{Kw}{input}, item $\backslash$SetKw{Kw}{thetext}, item $\backslash$SetKwData{Kw}{thetext}
  * $\backslash$KwA item 2, item 2
  * $\backslash$KwDados 1st item
  * $\backslash$KwData 3rd item
  * $\backslash$KwDati 1st item
  * $\backslash$KwEntrada 1st item
  * $\backslash$KwFn item $\backslash$SetKwFunction{KwFn}{Fn}
  * $\backslash$KwFunction item nokwfunc
  * $\backslash$KwIn 1st item
  * $\backslash$KwIng 1st item
  * $\backslash$KwOut 2nd item
  * $\backslash$KwResult 4th item
  * $\backslash$KwResultado 1st item
  * $\backslash$KwRet 1st item
  * $\backslash$KwRetorna 2nd item
  * $\backslash$KwRisult 1st item
  * $\backslash$KwRitorna 2nd item
  * $\backslash$KwSaida 1st item
  * $\backslash$KwSty item 7, item $\backslash$SetBlockMarkersSty{$<$font$>$}, item $\backslash$BlockMarkersSty{$<$text$>$}, item $\backslash$KwSty{$<$text$>$}, item $\backslash$KwSty{$<$text$>$}
  * $\backslash$KwTo 1st item
  * $\backslash$KwUsc 1st item
  * $\backslash$KwZurueck 2nd item
  * $\backslash$label §3.1, §5, item $\backslash$ShowLn{label}, item $\backslash$nl, item $\backslash$lnl{label}, item $\backslash$lnl{label}, item $\backslash$lnlset{text}{label}, §9.2.1
  * $\backslash$lAltrimenti 5th item
  * $\backslash$lAltSe 5th item
  * $\backslash$lAnderes 6th item
  * $\backslash$lAutre item 5
  * $\backslash$lCas item 5
  * $\backslash$lCase 7th item, 8th item, 11st item, 6th item
  * $\backslash$lCase(comment) 13rd item
  * $\backslash$lCase* 12nd item
  * $\backslash$lCaso item 5, 5th item
  * $\backslash$leftskip item $\backslash$algomargin
  * $\backslash$lElse 17th item, 18th item, 18th item
  * $\backslash$lElse(comment) 19th item
  * $\backslash$lElse* 20th item
  * $\backslash$lElseIf 11st item, 12nd item, 15th item
  * $\backslash$lElseIf(comment) 16th item
  * $\backslash$lElseIf* 17th item
  * $\backslash$lEnOtroCaso item 4
  * $\backslash$lEnOtroCasoSi item 4
  * $\backslash$lEnqto 10th item
  * $\backslash$lFall 6th item
  * $\backslash$lFinche 11st item
  * $\backslash$lFor 3rd item, 4th item, 5th item
  * $\backslash$lFor(comment) 6th item
  * $\backslash$lFor* 7th item
  * $\backslash$lForAll 3rd item, 4th item
  * $\backslash$lForEach 3rd item, 4th item
  * $\backslash$lFuer 7th item
  * $\backslash$lFuerAlle 10th item
  * $\backslash$lFuerJedes 9th item
  * $\backslash$lFuerPar 8th item
  * $\backslash$lIf 5th item, 6th item, 12nd item, 12nd item, 14th item
  * $\backslash$lIf(comment) 13rd item
  * $\backslash$lIf* 14th item
  * $\backslash$Line item $\backslash$SetKwBlock{Begin}{begin}{end}
  * lined item lined:, item lined:
  * $\backslash$lineskip item interspacetitleboxruled
  * $\backslash$LinesNotNumbered 16th item, item $\backslash$LinesNotNumbered, item $\backslash$LinesNotNumbered
  * $\backslash$linesnotnumbered 16th item
  * $\backslash$LinesNumbered 15th item, item $\backslash$LinesNumbered, item $\backslash$LinesNumbered
  * $\backslash$linesnumbered item $\backslash$SetKwHangingKw{Kw}{thetext}, 15th item, item linesnumbered:, item linesnumbered:, item linesnumberedhidden:, item $\backslash$LinesNumbered, §9.3.1, §9.7
  * $\backslash$LinesNumberedHidden 17th item, item $\backslash$LinesNumberedHidden, item $\backslash$LinesNumberedHidden
  * $\backslash$linesnumberedhidden 17th item, item linesnumberedhidden:, item linesnumberedhidden:, item $\backslash$LinesNumberedHidden, item $\backslash$ShowLn, §9.3.1
  * $\backslash$listofalgorithmes item algo2e:
  * $\backslash$listofalgorithms §2, item $\backslash$listofalgorithms, item $\backslash$listofalgorithms, §9.2.1
  * $\backslash$lMientras item 8
  * $\backslash$lnl item $\backslash$lnl{label}, item $\backslash$lnl{label}, item $\backslash$lnlset{text}{label}
  * $\backslash$lnlset item $\backslash$lnlset{text}{label}, item $\backslash$lnlset{text}{label}
  * longend item longend, item longend, item $\backslash$SetAlgoLongEnd
  * $\backslash$lOther 11st item, 12nd item, 18th item, 6th item
  * $\backslash$lOther(comment) 20th item
  * $\backslash$lOther* 19th item
  * $\backslash$lOtro item 5
  * $\backslash$lOutro 5th item
  * $\backslash$lPara item 6, 6th item
  * $\backslash$lParaCada 8th item
  * $\backslash$lParaPar 7th item
  * $\backslash$lParaPara item 7
  * $\backslash$lParaTodo item 10, 9th item
  * $\backslash$lPer 7th item
  * $\backslash$lPerCiascun 9th item
  * $\backslash$lPerPar 8th item
  * $\backslash$lPerTutti 10th item
  * $\backslash$lPour item 6
  * $\backslash$lPourCh item 8, item 9
  * $\backslash$lPourTous item 9
  * $\backslash$lRepeat 5th item, 6th item, 5th item
  * $\backslash$lRepeat(comment) 7th item
  * $\backslash$lRepeat* 6th item
  * $\backslash$lRepeter item 10
  * $\backslash$lRepetir item 11
  * $\backslash$lRepita 11st item
  * $\backslash$lRipeti 4th item
  * $\backslash$lSe 4th item
  * $\backslash$lSea 5th item
  * $\backslash$lSenao 4th item
  * $\backslash$lSenaoSe 4th item
  * $\backslash$lSi item 4
  * $\backslash$lSinon item 4
  * $\backslash$lSinonSi item 4
  * $\backslash$lSolange 11st item
  * $\backslash$lSonst 5th item
  * $\backslash$lSonstWenn 5th item
  * $\backslash$lSSi item 4
  * $\backslash$lTq item 7
  * $\backslash$lWenn 5th item
  * $\backslash$lWhile 3rd item, 4th item
  * $\backslash$lWiederh 4th item
  * $\backslash$Mientras item 8
  * ngerman item ngerman:, item ngerman:
  * $\backslash$nl §3.2, item $\backslash$nl, item $\backslash$nl, item $\backslash$lnl{label}, item $\backslash$nlset{text}, item $\backslash$lnlset{text}{label}
  * $\backslash$nllabel item linesnumbered:, item $\backslash$nllabel{label}, item $\backslash$nllabel{label}
  * $\backslash$nlset §3.2, item $\backslash$nlset{text}, item $\backslash$nlset{text}, item $\backslash$lnlset{text}{label}
  * $\backslash$NlSty 13rd item, item $\backslash$NlSty{$<$text$>$}, item $\backslash$NlSty{$<$text$>$}
  * $\backslash$nlSty 13rd item
  * noalgohanging item noalgohanging, item noalgohanging
  * $\backslash$NoCaptionOfAlgo 20th item, item $\backslash$NoCaptionOfAlgo, item $\backslash$NoCaptionOfAlgo, item $\backslash$RestoreCaptionOfAlgo
  * $\backslash$nocaptionofalgo 20th item
  * noend item noend, item noend, item $\backslash$SetAlgoNoEnd
  * nofillcomment item nofillcomment, item nofillcomment, item $\backslash$SetNoFillComment
  * nokwfunc 6th item, item nokwfunc, item nokwfunc
  * $\backslash$NoLine item $\backslash$SetInd{before rule space}{after rule space}
  * noline item noline:, item noline:
  * norelsize item article-hermes, item norelsize:, item norelsize:
  * noresetcount item noresetcount, item noresetcount
  * oldcommands §5
  * onelanguage item onelanguage:, item onelanguage:
  * opthanginginout item opthanginginout, item opthanginginout
  * $\backslash$Other 10th item, 9th item, 14th item, 16th item, 6th item
  * $\backslash$Other(comment) 15th item, 17th item
  * $\backslash$Otro item 5
  * $\backslash$Output item $\backslash$SetKwInput{Kw}{input}
  * $\backslash$Outro 5th item
  * $\backslash$Para item 6, 6th item
  * $\backslash$ParaCada item 9, 8th item
  * $\backslash$ParaPar 7th item
  * $\backslash$ParaPara item 7
  * $\backslash$ParaTodo item 10, 9th item
  * $\backslash$parindent item $\backslash$algomargin
  * $\backslash$Per 7th item
  * $\backslash$PerCiascun 9th item
  * $\backslash$PerPar 8th item
  * $\backslash$PerTutti 10th item
  * plain item plain:, item plain:, item $\backslash$RestyleAlgo{style}
  * portuguese item portuguese:, item portuguese:
  * portuguesekw item portuguesekw:, item portuguesekw:
  * $\backslash$Pour item 6
  * $\backslash$PourCh item 8
  * $\backslash$PourTous item 9
  * $\backslash$PrintSemicolon 6th item, item $\backslash$PrintSemicolon, item $\backslash$PrintSemicolon
  * $\backslash$printsemicolon 6th item
  * $\backslash$ProcArgFnt item 5, item $\backslash$ProcArgFnt{$<$text$>$}, item $\backslash$ProcArgFnt{$<$text$>$}, item $\backslash$ProcArgFnt{$<$text$>$}, item $\backslash$ProcArgFnt{$<$text$>$}, item $\backslash$ProcArgFnt{$<$text$>$}, item $\backslash$ProcArgSty{$<$text$>$}, item $\backslash$ProcArgSty{$<$text$>$}
  * $\backslash$ProcArgSty item 5, item $\backslash$ProcArgSty{$<$text$>$}, item $\backslash$ProcArgSty{$<$text$>$}, item $\backslash$ProcArgFnt{$<$text$>$}, item $\backslash$ProcArgFnt{$<$text$>$}, item $\backslash$ProcArgSty{$<$text$>$}, item $\backslash$ProcArgSty{$<$text$>$}, item $\backslash$ProcArgSty{$<$text$>$}, item $\backslash$ProcArgSty{$<$text$>$}, §9.5.2
  * $\backslash$ProcFnt item $\backslash$ProcFnt{$<$text$>$}, item $\backslash$ProcFnt{$<$text$>$}, item $\backslash$ProcArgFnt{$<$text$>$}, item $\backslash$ProcSty{$<$text$>$}, item $\backslash$ProcSty{$<$text$>$}, item $\backslash$ProcFnt{$<$text$>$}, item $\backslash$ProcFnt{$<$text$>$}, item $\backslash$ProcFnt{$<$text$>$}, item $\backslash$ProcFnt{$<$text$>$}, §9.5.2, §9.5.2, §9.5.2, §9.5.2
  * $\backslash$ProcNameFnt item 4, item $\backslash$ProcNameSty{$<$text$>$}, item $\backslash$ProcNameSty{$<$text$>$}, item $\backslash$ProcNameFnt{$<$text$>$}, item $\backslash$ProcNameFnt{$<$text$>$}, item $\backslash$ProcNameFnt{$<$text$>$}, item $\backslash$ProcNameFnt{$<$text$>$}, §9.5.2, §9.5.2, §9.5.2, §9.5.2, §9.5.2
  * $\backslash$ProcNameSty item 4, item $\backslash$ProcNameSty{$<$text$>$}, item $\backslash$ProcNameSty{$<$text$>$}, item $\backslash$ProcNameSty{$<$text$>$}, item $\backslash$ProcNameSty{$<$text$>$}, item $\backslash$ProcNameSty{$<$text$>$}, item $\backslash$ProcNameSty{$<$text$>$}, item $\backslash$ProcNameFnt{$<$text$>$}, item $\backslash$ProcNameFnt{$<$text$>$}, §9.5.2, §9.5.2, §9.5.2, §9.5.2, §9.5.2
  * procnumbered item procnumbered:, item procnumbered:
  * $\backslash$ProcSty item $\backslash$ProcSty{$<$text$>$}, item $\backslash$ProcSty{$<$text$>$}, item $\backslash$ProcSty{$<$text$>$}, item $\backslash$ProcSty{$<$text$>$}, item $\backslash$ProcFnt{$<$text$>$}, item $\backslash$ProcFnt{$<$text$>$}, §9.5.2, §9.5.2, §9.5.2, §9.5.2
  * $\backslash$ProgArgFnt §9.5.2, §9.5.2, §9.5.2
  * $\backslash$ProgArgSty §9.5.2, §9.5.2
  * $\backslash$ProgSty item $\backslash$ProgSty{$<$text$>$}, item $\backslash$ProgSty{$<$text$>$}
  * pstcol 2nd item
  * $\backslash$ref §3.2, item $\backslash$nlset{text}
  * $\backslash$relax item $\backslash$SetAlCapFnt{$<$font$>$}, item $\backslash$SetAlCapNameFnt{$<$font$>$}, item $\backslash$AlCapFnt{$<$text$>$}, item $\backslash$AlCapNameFnt{$<$text$>$}, item $\backslash$ProcFnt{$<$text$>$}, item $\backslash$ProcNameFnt{$<$text$>$}
  * $\backslash$Repeat 1st item, 2nd item, 3rd item, 4th item, 1st item, 3rd item
  * $\backslash$Repeat(comment after repeat) 2nd item, 4th item
  * $\backslash$Repeter item 10
  * $\backslash$Repetir item 11
  * $\backslash$Repita 11st item
  * $\backslash$Res item 1
  * resetcount item resetcount, item resetcount
  * $\backslash$ResetInOut item $\backslash$SetKwInOut{Kw}{input}
  * $\backslash$RestoreCaptionOfAlgo 21st item, item $\backslash$RestoreCaptionOfAlgo, item $\backslash$RestoreCaptionOfAlgo
  * $\backslash$restorecaptionofalgo 21st item
  * $\backslash$RestyleAlgo 22nd item, item $\backslash$RestyleAlgo{style}, item $\backslash$RestyleAlgo{style}, item $\backslash$RestyleAlgo{style}, item $\backslash$RestyleAlgo{style}
  * $\backslash$restylealgo 22nd item
  * $\backslash$Resultado item 1
  * $\backslash$Retorna 2nd item
  * $\backslash$Retour item 2
  * $\backslash$Return 2nd item
  * rightnl item rightnl:, item rightnl:
  * $\backslash$Ripeti 4th item
  * $\backslash$Ritorna 2nd item
  * ruled §3.2, item ruled:, item ruled:, item $\backslash$RestyleAlgo{style}, item $\backslash$NoCaptionOfAlgo
  * $\backslash$Salida item 1
  * scleft item scleft, item scleft, item $\backslash$SetSideCommentLeft
  * scright item scright (default), item scright (default), item $\backslash$SetSideCommentRight
  * $\backslash$Se 4th item
  * $\backslash$Sea 5th item
  * $\backslash$Selec 5th item
  * $\backslash$Seleccionar item 5
  * $\backslash$Senao 4th item
  * $\backslash$SetAlCapFnt item $\backslash$SetAlCapFnt{$<$font$>$}, item $\backslash$SetAlCapFnt{$<$font$>$}, §9.5.2, §9.5.3, §9.5.3, §9.5.4
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# A study of topological structures on equi-continuous mappings

Ankit Gupta1 and Ratna Dev Sarma2 1Department of Mathematics, University of
Delhi, Delhi 110007, India.
Email<EMAIL_ADDRESS>
2Department of Mathematics, Rajdhani College $($University of Delhi$)$, Delhi
110015, India.
Email<EMAIL_ADDRESS>
###### Abstract.

Function space topologies are developed for $EC(Y,Z)$, the class of equi-
continuous mappings from a topological space $Y$ to a uniform space $Z$.
Properties such as splittingness, admissibility etc. are defined for such
spaces. The net theoretic investigations are carried out to provide
characterizations of splittingness and admissibility of function spaces on
$EC(Y,Z)$. The open-entourage topology and point-transitive-entourage topology
are shown to be admissible and splitting respectively. Dual topologies are
defined. A topology on $EC(Y,Z)$ is found to be admissible (resp. splitting)
if and only if its dual is so.

###### Key words and phrases:

topology; uniform space; function Space; equi-continuous mappings; net
convergence

###### 2010 Mathematics Subject Classification:

54C35; 54A05

Corresponding author $:$ Ankit Gupta
This paper was prepared during a sabbatical leave of the second author.

##  1\. Introduction

Investigations of topological aspects of the collections of continuous
mappings from a topological space $Y$ to another topological space $Z$ has
been an area of active research in topology. Intrinsic properties of function
space topologies have also been investigated in depth by several researchers.
The relationship between convergence and topologies of
$\mathcal{C}(X,\mathbb{R})$ and that of the hyperspaces $\mathcal{C}(X,\$)$ of
open subsets of $X$ has been studied in [4]. Dual topologies for function
space topologies and existence of a greatest splitting topology have been
investigated in [5] and [6] respectively. Conditions under which compact-open,
Isbell or natural topologies etc. on $\mathcal{C}(X,\mathbb{R})$ may coincide
have been explored in [9]. In the recent years, several research papers have
come up dealing with certain particular as well as some more general cases of
this study. For example, for the particular case $Z=\mathbb{R}$, bounded-open
topology and pseudo-compact-open topologies are discussed in [13] and [14]. In
[2] and [3], function space topologies arising from strong uniform continuity
have been studied. On the other hand in [10] and [11], topologies on $Y$ and
$Z$ are replaced by fuzzy topologies, which provide a more general set up for
topological properties. Similarly, function space topologies for generalized
topological spaces have been discussed in [7]. In this present paper, we
investigate the same for equi-continuous mappings from $Y$ to $Z$, where $Y$
has a topology while $Z$ is equipped with a uniformity. With the help of
examples, we have shown that several such topologies do exist really on
$EC(Y,Z)$, the collection of equi-continuous mappings from $Y$ to $Z$. As the
uniform spaces are positioned between the metric spaces and the topological
spaces, there is a tendency to discount their investigations as particular
cases of topology. However, through our study, we have shown here that uniform
structures and in particular, the equi-continuous mappings need not to be
studied from that point of view. Rather the inherent aesthetics and
intricacies, arising out of uniformities are best revealed, when the related
notions are studied directly, not as by product of topology. In fact, we have
also introduced function space topology for the family of ${}_{p}EC(Y,Z)$ of
pseudo-dislocated equi-continuous mappings. In this case, $Z$ has pseudo-
dislocated uniformity, which unlike uniformity, does not generate any
topology.

We have introduced admissibility and splittingness for $EC(Y,Z)-$ two
important features for any function space topology. Using net-theory, we have
developed the concept of equi-continuous convergence of nets of equi-
continuous functions. Splittingness and admissibility are characterized using
the notion of equi-continuous convergence. These characterizations are used to
prove that open-entourage topology on $EC(Y,Z)$ is admissible whereas point-
transitive-entourage topology is splitting. In the last section, we have
introduced the concept of dual topology on $\mathcal{O}_{Z}(Y)$, the
collection of open sets of $Y$ obtained in relation to the equi-continuous
mappings. Interesting relationships are observed between the topologies on
$\mathcal{O}_{Z}(Y)$ and that of $EC(Y,Z)$. For example, a topology on
$EC(Y,Z)$ is admissible (resp. splitting) if and only if its dual on
$\mathcal{O}_{Z}(Y)$ is admissible (resp. splitting). Similarly, a topology on
$\mathcal{O}_{Z}(Y)$ is admissible (resp. splitting) if and only if its dual
on $EC(Y,Z)$ is so.

##  2\. Equi-Continuity, Pseudo-dislocated equi-continuity and Convergence

In this section, we develop the net convergence criterion for equi-
continuousas well as pseudo-dislocated-equi-continuity mappings.

######  Definition 2.1.

A uniform structure or uniformity on a non-empty set $X$ is a family
$\mathcal{U}$ of subsets of $X\times X$ satisfying following properties:

  1. $(\mathcal{U}_{1})$

if $U\in\mathcal{U}$, then $\Delta X\in U$;
here $\Delta X=\\{(x,x)\in X\times X$ for all $x\in X\\}$;

  2. $(\mathcal{U}_{2})$

if $U\in\mathcal{U}$, then $U^{-1}\in\mathcal{U}$;
here, $U^{-1}$ is called inverse relation of $U$ and defined as

$U^{-1}=\\{(x,y)\in X\times X\,|\,(y,x)\in U\\}$

  3. $(\mathcal{U}_{3})$

if $U\in\mathcal{U}$, then there exists some $V\in\mathcal{U}$ such that
$V\circ V\subseteq U$;
here the composition $U\circ V=\\{(x,z)\in X\times X\,|\,$ for some $y\in X$,
$(x,y)\in V$ and $(y,z)\in U\\}$.

  4. $(\mathcal{U}_{4})$

if $U,V\in\mathcal{U}$, then $U\cap V\in\mathcal{U}$;

  5. $(\mathcal{U}_{5})$

if $U\in\mathcal{U}$ and $U\subseteq V\subseteq X\times X$, then
$V\in\mathcal{U}$.

The pair $(X,\mathcal{U})$ is a uniform space and the members of $\mathcal{U}$
are called entourages.

######  Definition 2.2.

Let $(X,\tau)$ and $(Y,\mathcal{U})$ be a topological space and a uniform
space respectively. A function $f\,:(X,\tau)\to(Y,\mathcal{U})$ is said to be
equi-continuous at $x\in X$, if for each entourage $U\in\mathcal{U}$, there
exists an open neighbourhood $V$ of $x$ such that $f(V)\subseteq U[f(x)]$,
where $U[f(x)]=\\{y\in Y\,|\,((f(x),y))\in U\\}$.

If $f$ is equi-continuous for all $x\in X$, then $f$ is called equi-
continuous.
The collection of all equi-continuous functions from $X$ to $Y$ is denoted by
$EC(X,Y)$ respectively.

######  Definition 2.3.

Let $\\{x_{n}\\}_{n\in D}$ be a net in a uniform space $(X,\mathcal{U})$ .
Then $\\{x_{n}\\}_{n\in D}$ is said to be convergent to $x\in X$, if for each
entourage $U\in\mathcal{U}$, there exists an $m\in D$, such that $(x,x_{n})\in
U$ for all $n\geq m$.

In our next theorem, we provide the net convergence criteria for equi-
continuous functions.

######  Theorem 2.4.

Let $(X,\tau)$ and $(Y,\mathcal{U})$ be a topological space and a uniform
space respectively. Then a function $f\,:(X,\tau)\to(Y,\mathcal{U})$ is equi-
continuous at $x\in X$ if and only if whenever a net $\\{x_{n}\\}_{n\in D}$
converges to $x$ in $X$, its image net $\\{f(x_{n})\\}_{n\in D}$ converges to
$f(x)$ in $Y$.

  * Proof.

Let $\\{x_{n}\\}_{n\in D}$ be any convergent net in $X$, which converges to
$x\in X$ and let $f\,:X\to Y$ be equi-continuous at $x\in X$. We have to show
that the net $\\{f(x_{n})\\}_{n\in D}$ converges to $f(x)$ in $Y$. Let
$U\in\mathcal{U}$ be any entourage. Since $f$ is equi-continuous at $x\in X$,
therefore there exists an open neighbourhood $V$ of $x$ such that
$f(V)\subseteq U[f(x)]$. Since the net $\\{x_{n}\\}_{n\in D}$ converges to
$x$, $x_{n}\in V$ eventually. Hence $f(x_{n})\in f(V)\subseteq U[f(x)]$
eventually which implies that $(f(x),f(x_{n}))\in U$ eventually. Therefore the
image net $\\{f(x_{n})\\}_{n\in D}$ converges to $f(x)$ in $Y$.
Conversely, let the hypothesis hold. Let if possible $f$ be not equi-
continuous at $x\in X$. Then there exists an entourage $U\in\mathcal{U}$ such
that there is no open neighbourhood $V$ of $x\in X$ such that $f(V)\subseteq
U[f(x)]$. That is, for each open neighbourhood $V$ of $x$, there exists some
$x_{V}\in V$ such that $f(x_{V})\notin U[f(x)]$. Let $D$ be a collection of
all open neighbourhoods of $x$. Then $(D,\geq)$ is a directed set under the
inverse set inclusion $\geq$, that is, $V\geq U$ if $V\subseteq U$. Then
$\\{x_{V}\\}_{V\in D}$ is a net in $X$ which converges to $x$. But the image
net $\\{f(x_{V})\\}_{V\in D}$ does not converge to $f(x)$, because for
$U\in\mathcal{U}$, we have $f(x_{V})\notin U[f(x)]$ for all $V\in D$. Thus we
get a contradiction. Therefore $f$ is equi-continuous at $x\in X$. ∎

Next we provide few results regarding the pseudo-dislocated uniform space and
pseudo-dislocated equi-continuous mappings. The importance of these spaces
lies in the fact that they do not generate any topology like the uniform
spaces do.

######  Definition 2.5.

[12] A pseudo-dislocated uniformity on a non-empty set $X$ associated with a
subset $A$ of $X$ is a family $\mathcal{U}_{A}$ of subsets of $X\times X$
which satisfies
$(\mathcal{U}_{2}),(\mathcal{U}_{3}),(\mathcal{U}_{4}),(\mathcal{U}_{5})$ of
Definition 2.1 together with the following property:
($\mathcal{U}_{1}^{\prime})\;$ Every member of $\mathcal{U}_{A}$ contains
$\Delta_{A}=\\{(x,x)\,|\,x\in A\\}$.

The pair $(X,\mathcal{U}_{A})$ is called pseudo-dislocated uniform space.

######  Definition 2.6.

Let $(X,\tau)$ and $(Y,\mathcal{U}_{A})$ be a topological space and a pseudo-
dislocated uniform space respectively. A function
$f\,:(X,\tau)\to(Y,\mathcal{U}_{A})$ is said to be pseudo-dislocated equi-
continuous at $x\in X$, if for each entourage $U\in\mathcal{U}_{A}$, there
exists an open neighbourhood $V$ of $x$ such that $f(V)\subseteq U[f(x)]$,
where $U[f(x)]=\\{y\in Y\,|\,((f(x),y))\in U\\}$.

If $f$ is pseudo-dislocated equi-continuous for all $x\in X$, then $f$ is
called pseudo-dislocated equi-continuous and the collection of all pseudo-
dislocated equi-continuous functions from $X$ to $Y$ is denoted by
${}_{p}EC(X,Y)$ respectively.

######  Definition 2.7.

Let $\\{x_{n}\\}_{n\in D}$ be a net in a pseudo-dislocated uniform space
$(X,\mathcal{U}_{A})$ . Then $\\{x_{n}\\}_{n\in D}$ is said to be convergent
to $x\in X$, if for each entourage $U\in\mathcal{U}_{A}$, there exists an
$m\in D$, such that $(x,x_{n})\in U$ for all $n\geq m$.

We can show that the following net convergence criteria result holds good for
pseudo-dislocated equi-continuous mappings

######  Theorem 2.8.

Let $(X,\tau)$ and $(Y,\mathcal{U}_{A})$ be a topological space and a pseudo-
dislocated uniform space respectively. Then a function
$f\,:(X,\tau)\to(Y,\mathcal{U}_{A})$ is pseudo-dislocated equi-continuous at
$x\in X$ if and only if whenever a net $\\{x_{n}\\}_{n\in D}$ converges to $x$
in $X$, its image net $\\{f(x_{n})\\}_{n\in D}$ converges to $f(x)$ in $Y$.

##  3\. Topologies on $EC(Y,Z)$

In this section, we introduce few topologies on $EC(Y,Z)$ and ${}_{p}EC(Y,Z)$.
Let $(Y,\tau)$ and $(Z,\mathcal{U})$ be a topological space and a uniform
space respectively. Then for $z\in Z$, $V\in\tau$ and $U\in\mathcal{U}$, we
define:

$(V,U)_{z}=\\{f\in EC(Y,Z)\,|\,f(V)\subseteq U[z]\,\,\\}$

Let $\mathcal{S}_{\tau,\mathcal{U}}=\\{(V,U)_{z}\,|\,z\in Z,\,\,V\in\tau\text{
and }U\in\mathcal{U}\\}$.

######  Lemma 3.1.

$\mathcal{S}_{\tau,\mathcal{U}}$ is a subbasis for a topology on $EC(Y,Z)$.

  * Proof.

Let $f\in EC(Y,Z)$. Then for $y\in Y$ and $U\in\mathcal{U}$, there exists some
open neighbourhood $V_{0}$ of $y\in Y$ such that $f(V_{0})\subseteq U[f(y)]$.
Consider $f(y)=z\in Z$. Then we have $f\in(V_{0},U)_{z}$. Therefore
$EC(Y,Z)\subseteq\bigcup\mathcal{S}_{\tau,\mathcal{U}}$. ∎

The topology generated by this subbasis will be called the open-entourage
topology for $EC(Y,Z).$

Similarly, for $y\in Y$, $V\in\tau$ and $U\in\mathcal{U}$, let us consider
$(V,U)_{y}=\\{f\in EC(Y,Z)\,|\,f(V)\subseteq U[f(y)]\,\,\\}$. Then it can be
shown that the collection $\\{(V,U)_{y}\,|\,y\in Y,V\in\tau\text{ and
}U\in\mathcal{U}\\}$ also forms a subbasis for a topology on $EC(Y,Z)$.
The topology generated by this subbasis is called the open-entourage topology
of Type-I for $EC(Y,Z)$.

Clearly, the open-entourage topology is finer than the open-entourage topology
of Type-I.
Similarly, let $(Y,\tau)$ and $(Z,\mathcal{U})$ be a topological space and a
uniform space respectively and $y\in Y$, $z\in Z$ and for $U\in\mathcal{U}$.
We define:

$(y,U)_{z}=\\{f\in EC(Y,Z)\,|\,f(y)\in U[z]\,\,\\}$.

Let $\mathcal{S}_{\tau,\mathcal{U}}^{p}=\\{(y,U)_{z}\,|\,z\in Z,\,\,y\in
Y\text{ and }U\in\mathcal{U}\\}$.

######  Lemma 3.2.

$\mathcal{S}_{\tau,\mathcal{U}}^{p}$ is a subbasis for a topology on
$EC(Y,Z)$.

  * Proof.

Let $f\in EC(Y,Z)$, then for $y\in Y$ and for $U\in\mathcal{U}$ we have
$\left(f(y),f(y)\right)\in U$. Consider $f(y)=z\in Z$, then we have $f(y)\in
U[z]$. Hence $f\in(y,U)_{z}$ and therefore
$EC(Y,Z)\subseteq\bigcup\mathcal{S}_{\tau,\mathcal{U}}^{p}$. ∎

The topology generated by this subbasis will be called the point-entourage
topology for $EC(Y,Z).$
Now, for $y\in Y$, $z\in Z$. Let $U_{t}\in\mathcal{U}$ be a transitive
entourage of $\mathcal{U}$, that is $U_{t}\circ U_{t}\subset U_{t}$.
We define:

$(y,U_{t})_{z}=\\{f\in EC(Y,Z)\,|\,f(y)\in U_{t}[z]\,\,\\}$.

For each uniform space $(Z,\mathcal{U})$, we have $Z\times Z\in\mathcal{U}$.
Then $U_{t}=Z\times Z$ satisfies the property $U_{t}\circ U_{t}\subseteq
U_{t}$. Therefore there always exists entourages of the type
$U_{t}\in\mathcal{U}$.
Let $\mathcal{S}_{\tau,\mathcal{U}}^{t,p}=\\{(y,U_{t})_{z}\,|\,z\in Z,\,\,y\in
Y,U_{t}\in\mathcal{U}\text{ such that }U_{t}\circ U_{t}\subseteq U_{t}\\}$.
It may be verified that $\mathcal{S}_{\tau,\mathcal{U}}^{t,p}$ is a subbasis
for a topology on $EC(Y,Z)$.
The topology generated by this subbasis will be called the point-transitive-
entourage topology for $EC(Y,Z).$
Now, we introduce a topological structure on the class of pseudo-dislocated
equi-continuous function ${}_{p}EC(Y,Z)$.

Let $(Y,\tau)$ and $(Z,\mathcal{U}_{A})$ be a topological space and a pseudo-
dislocated uniform space respectively. Then for $z\in Z$, $V\in\tau$ and
$U\in\mathcal{U}_{A}$, we define

$(V,U)_{z}=\\{f\in_{p}EC(Y,Z)\,|\,f(V)\subseteq U[z]\\}$

Let $\mathcal{S}(\tau,\mathcal{U}_{A})=\\{(V,U)_{z}\,|\,z\in Z,V\in\tau\text{
and }U\in\mathcal{U}_{A}\\}$.

######  Lemma 3.3.

$\mathcal{S}(\tau,\mathcal{U}_{A})$ is a subbasis for a topology on
${}_{p}EC(Y,Z)$.

  * Proof.

Similar to Lemma 3.1. ∎

The topology generated by this subbasis will be called the open-dislocated-
entourage for ${}_{p}EC(Y,Z)$.

In the following section, we provide investigations of the function spaces on
$EC(Y,Z)$. The development for ${}_{p}EC(Y,Z)$, being similar, is not shown in
the paper to avoid repetition.

##  4\. Admissibility and Splittingness on $EC(Y,Z)$

In this section, we introduce few topologies on $EC(Y,Z)$ and investigate some
of their properties. Admissibility and splittingness for such spaces are
defined and their characterizations are also provided in this section.

######  Definition 4.1.

Let $(Y,\tau)$ and $(Z,\mathcal{U})$ be a topological space and a uniform
space respectively. Let $(X,\mu)$ be another topological space. Then for a map
$g\,:X\times Y\to Z$, we define a map $g^{*}\,:X\to EC(Y,Z)$ by
$g^{*}(x)(y)=g(x,y)$.

These mappings $g$ and $g^{*}$ are called associated maps.

######  Definition 4.2.

Let $(Y,\tau)$ and $(Z,\mathcal{U})$ be a topological space and a uniform
space respectively. A topology $\mathfrak{T}$ on $EC(Y,Z)$ is called

  1. $(i)$

admissible if the evaluation map $e\,:EC(Y,Z)\times Y\to Z$ defined by
$e(f,y)=f(y)$ is equi-continuous.

  2. $(ii)$

splitting if for each topological space $(X,\mu)$, equi-continuity of the map
$g\,:X\times Y\to Z$ implies continuity of the map $g^{*}\,:X\to EC(Y,Z)$,
where $g^{*}$ is the associated map of $g$.

The following results show that equi-continuity at times behaves like
continuity only.

######  Proposition 4.3.

Let $(X,\tau)$ and $(Y,\mu)$ be two topological spaces and $(Z,\mathcal{U})$
be a uniform space. Let $f\,:X\to Y$ and $g\,:Y\to Z$ be continuous and equi-
continuous functions at $x\in X$ and $f(x)\in Y$ respectively. Then the
composition map $g\circ f\,:X\to Z$ is equi-continuous at $x\in X$.

  * Proof.

Let $U\in\mathcal{U}$ be any entourage in $\mathcal{U}$. Since the map $g$ is
equi-continuous at $f(x)$, therefore there exists an open neighbourhood $V$ of
$f(x)$ in $Y$, such that $g(V)\subseteq U[g(f(x))]$. We have $f(x)\in V$ and
$f$ is continuous at $x$, thus there exists an open neighbourhood $W$ of $x$
in $X$ with $f(W)\subseteq V$. Hence, we have $g(f(W))\subseteq g(V)\subseteq
U[g(f(x))]$, that is, $(g\circ f)(W)\subseteq U[(g\circ f)(x)]$. Therefore the
composition map $g\circ f$ is equi-continuous at $x$. ∎

In the light of the above result, now we provide a characterization of
admissibility.

######  Theorem 4.4.

Let $(Y,\tau)$ and $(Z,\mathcal{U})$ be a topological space and a uniform
space respectively. Let $(X,\mu)$ be any topological space. Then a topology
$\mathfrak{T}$ on $EC(Y,Z)$ is admissible if and only if continuity of the map
$g^{*}\,:X\to EC(Y,Z)$ implies equi-continuity of the map $g\,:X\times Y\to
Z$, where $g^{*}$ and $g$ are the associated maps.

  * Proof.

Let the topology $\mathfrak{T}$ on $EC(Y,Z)$ be admissible, that is, the
evaluation map $e\,:EC(Y,Z)\times Y\to Z$ be equi-continuous. Let
$g^{*}\,:X\to EC(Y,Z)$ be any continuous map. We have to show that its
associated map $g$ is equi-continuous. Since the map $g^{*}$ is continuous,
therefore the map $h\,:X\times Y\to EC(Y,Z)\times Y$, defined by
$h(x,y)=(g^{*}(x),y)$ is also continuous. Hence, by the last proposition, the
composition map $e\circ h$ is equi-continuous. Now, for $(x,y)\in X\times Y$,
consider $e\circ h(x,y)=e(h(x,y))=e(g^{*}(x),y)=g^{*}(x)(y)=g(x,y)$. Hence
$e\circ h\equiv g$. Therefore the map $g$ is equi-continuous.
Conversely, let the condition hold. Consider $X=EC(Y,Z)$ with the topology
$\mathfrak{T}$. We define $g^{*}\,:EC(Y,Z)\to EC(Y,Z)$ as the identity map.
Hence $g^{*}$ is continuous. Thus by the given hypothesis, its associated map
$g\,:EC(Y,Z)\times Y\to Z$ is also equi-continuous. For any $(f,y)\in
EC(Y,Z)\times Y$, consider $g(f,y)=g^{*}(f)(y)=f(y)=e(f,y)$, where $e$ is the
evaluation map. Therefore $g\equiv e$ and hence equi-continuous. Thus the
topology $\mathfrak{T}$ on $EC(Y,Z)$ is admissible. ∎

In the next set of theorems, we provide characterizations of admissibility and
splittingness of the topologies on $EC(Y,Z)$ using net theory. We extend the
concept of continuous convergence of continuous mappings [1] for this purpose.
But before that we quote a result about directed sets, which we shall use in
our proof.
Let $\Delta$ be a directed set. We add a point $\infty$ to $\Delta$ satisfying
$\infty\geq n$ for all $n\in\Delta$ and write
$\Delta_{0}=\Delta\cup\\{\infty\\}$. A topology $\mathcal{T}_{0}$ may be
generated on $\Delta_{0}$ by declaring every singleton of $\Delta$ as open and
neighbourhoods of $\infty$ being of the form $U_{n_{0}}=\\{n\,\,:\,\,n\geq
n_{0}\\}$, $n_{0}\in\Delta$.

######  Lemma 4.5.

[8] Let $(Y,\tau)$ be a topological space and $\\{y_{n}\\}_{n\in D}$ be a net
in $Y$. Then the net $\\{y_{n}\\}_{n\in\Delta}$ converges to $y$ in $Y$ if and
only if the function $f\,:\Delta_{0}\to Y$ defined by $f(n)=y_{n}$ for
$n\in\Delta$ and $f(\infty)=y$ is continuous at $\infty$.

From this lemma we have the following remark:

######  Remark.

Let $(Y,\tau)$ be a topological space and $\\{y_{n}\\}_{n\in D}$ be net in
$Y$. Then the net $\\{y_{n}\\}_{n\in\Delta}$ converges to $y$ in $Y$ if and
only if the function $f\,:\Delta_{0}\to Y$ defined by $f(n)=y_{n}$ for
$n\in\Delta$ and $f(\infty)=y$ is continuous.

Now we come to our main results of this section.

######  Definition 4.6.

Let $\\{f_{n}\\}_{n\in\Delta}$ be a net in $EC(Y,Z)$. Then
$\\{f_{n}\\}_{n\in\Delta}$ is said to equi-continuously converge to $f\in
EC(Y,Z)$ if for each net $\\{y_{m}\\}_{m\in\sigma}$ in $Y$ converging to $y$,
$\\{f_{n}(y_{m})\\}_{(n,m)\in\Delta\times\sigma}$ converges to $f(y)$ in $Z$.

######  Theorem 4.7.

Let $(Y,\tau)$ and $(Z,\mathcal{U})$ be a topological space and a uniform
space respectively. Let $(X,\mu)$ be any topological space. Then a topology
$\mathfrak{T}$ on $EC(Y,Z)$ is splitting if and only if for each net
$\\{f_{n}\\}_{n\in\Delta}$ in $EC(Y,Z)$, equi-continuous convergence of
$\\{f_{n}\\}_{n\in\Delta}$ to $f$ implies that $\\{f_{n}\\}_{n\in\Delta}$
converges to $f$ under $\mathfrak{T}$.

  * Proof.

Let $\mathfrak{T}$ be splitting and $\\{f_{n}\\}_{n\in\Delta}$ equi-
continuously converge to $f$. Let $\Delta_{0}=\Delta\cup\\{\infty\\}$ be
equipped with the topology as described after Theorem 4.4. Define $g$ $:$
$\Delta_{0}\times Y\to Z$ by $g(n,y)=f_{n}(y)$ for all $n\in\Delta$ and
$g(\infty,y)=f(y)$. We show that the map $g$ is equi-continuous. Now, the only
non-constant convergent net in $\Delta_{0}$ is $\\{n\\}_{n\in\Delta}$ which
converges to $\infty$. Hence if $\mathcal{S}$ is a convergent net in
$\Delta_{0}\times Y$, then $\mathcal{S}=S_{1}\times S_{2}$, where
$S_{1}=\\{n\\}$ and $S_{2}=\\{y_{m}\\}_{m\in\sigma}$, where
$\\{y_{m}\\}_{m\in\sigma}$ is any convergent net in $Y$, which converges to
some $y$ in $Y$. Then $\mathcal{S}$ converges to $\\{\infty\\}\times\\{y\\}$
for some $y\in Y$ and
$g(\mathcal{S})=\\{f_{n}(y_{m})\\}_{(n,m)\in\Delta\times\sigma}$. By equi-
continuous convergence of $\\{f_{n}\\}_{n\in\Delta}$, $g(\mathcal{S})$
converges to $f(y)=g(\infty,y)$. Hence, by the net theoretic characterization
of equi-continuity, $g$ is equi-continuous at $(\infty,y)$. Now, consider any
$(n,y)\in\Delta\times Y$, and let $U$ be any entourage in $\mathcal{U}$. We
have, $g(n,y)=f_{n}(y)$. Since $U\in\mathcal{U}$ and $f_{n}$ is equi-
continuous, there exists an open neighbourhood $V$ of $y$ such that
$f_{n}(V)\subseteq U[f_{n}(y)]$. Thus, we get an open neighbourhood
$\\{n\\}\times V$ of $(n,y)$ such that $g(\\{n\\}\times V)=f_{n}(V)\subseteq
U[f_{n}(y)]$. That is, $g(\\{n\\}\times V)\subseteq U[g(n,y)]$. Therefore the
map $g$ is equi-continuous at $(n,y)$, for all $(n,y)\in\Delta\times Y$. As
$\mathfrak{T}$ is splitting, this implies that the associated map
$g^{*}\,:\Delta_{0}\to EC(Y,Z)$ is continuous. Since $\\{n\\}_{n\in\Delta}$
converges to $\infty$ in $\Delta_{0}$, we have, $\\{g^{*}(n)\\}_{n\in\Delta}$
converges to $g^{*}(\infty)$. Now $g^{*}(n)(y)=g(n,y)=f_{n}(y)$ and
$g^{*}(\infty)(y)=g(\infty,y)=f(y)$. That is, $g^{*}(n)=f_{n}$,
$g^{*}(\infty)=f$. Hence $\\{f_{n}\\}_{n\in\Delta}$ converges to $f$ in
$EC(Y,Z)$.

Conversely, suppose equi-continuous convergence implies convergence. Let $g$
$:$ $X\times Y\to Z$ be equi-continuous. We need to show that its associated
map $g^{*}$ is continuous. Let $\\{x_{n}\\}_{n\in\Delta}$ be any convergent
net in $X$ which converges to $x\in X$. We have to show that the image net
$\\{g^{*}(x_{n})\\}_{n\in\Delta}$ converges to $g^{*}(x)$ in $EC(Y,Z)$. We
define, $g^{*}(x_{n})=f_{n}$ and $g^{*}(x)=f$. Now, we show that
$\\{f_{n}\\}_{n\in\Delta}$ converges to $f$ in $EC(Y,Z)$. This follows if the
net $\\{f_{n}\\}_{n\in\Delta}$ equi-continuously converges to $f$. Let us
consider, a net $\\{y_{m}\\}_{m\in\sigma}$ in $Y$ which converges to some $y$
in $Y$. Then $\\{(x_{n},y_{m})\\}_{(n,m)\in\Delta\times\sigma}$ converges to
$(x,y)$ in $X\times Y$. As $g$ is equi-continuous, the image net
$\\{g(x_{n},y_{m})\\}_{(n,m)\in\Delta\times\sigma}$ converges to $g(x,y)$ in
$Z$. But $g(x_{n},y_{m})=g^{*}(x_{n})(y_{m})=f_{n}(y_{m})$ and
$g(x,y)=g^{*}(x)(y)=f(y)$. That is,
$\\{f_{n}(y_{m})\\}_{(n,m)\in\Delta\times\sigma}$ converges to $f(y)$ in $Z$.
Hence $\\{f_{n}\\}_{n\in\Delta}$ equi-continuously converges to $f$ in
$EC(Y,Z)$. Thus by the hypothesis, we have $\\{f_{n}\\}_{n\in\Delta}$
converges to $f$ in $EC(Y,Z)$. That is, $\\{g^{*}(x_{n})\\}_{n\in\Delta}$
converges to $g^{*}(x)$ in $EC(Y,Z)$. Hence $g^{*}$ is continuous. Therefore,
$\mathfrak{T}$ is splitting. ∎

On a similar line, characterization of admissibility is also provided below.

######  Theorem 4.8.

Let $(Y,\tau)$ and $(Z,\mathcal{U})$ be a topological space and a uniform
space respectively. Let $(X,\mu)$ be any topological space. Then a topology
$\mathfrak{T}$ on $EC(Y,Z)$ is admissible if and only if for each net
$\\{f_{n}\\}_{n\in\Delta}$ in $EC(Y,Z)$, convergence of
$\\{f_{n}\\}_{n\in\Delta}$ to $f$ in $EC(Y,Z)$ implies equi-continuous
convergence of $\\{f_{n}\\}_{n\in\Delta}$ to $f$.

  * Proof.

Let $\mathfrak{T}$ be admissible and $\\{f_{n}\\}_{n\in\Delta}$ be any net in
$EC(Y,Z)$ such that $\\{f_{n}\\}_{n\in\Delta}$ converges to $f$. Let us define
$g^{*}:\Delta_{0}\to EC(Y,Z)$ as $g^{*}(n)=f_{n}$ and $g^{*}(\infty)=f$, where
$\Delta_{0}$ is generated by $\Delta$. Now the only non constant convergent
net in $\Delta_{0}$ is $\\{n\\}$ which converges to $\infty$ and
$\\{g^{*}(n)\\}_{n\in\Delta}=f_{n}$ converges to $f=g^{*}(\infty)$, by the
given hypothesis. Hence $g^{*}$ is continuous. Therefore the associated map
$g:\Delta_{0}\times Y\to Z$ is equi-continuous. Let $\\{y_{m}\\}_{m\in\sigma}$
be any net in $Y$ such that $\\{y_{m}\\}_{m\in\sigma}$ converges to $y$ in
$Y$. Then $\\{(n,y_{m})\\}_{(n,m)\in\Delta\times\sigma}$ is a convergent net
in $\Delta_{0}\times Y$ which converges to $(\infty,y)$. Therefore
$\\{g(n,y_{m})\\}_{(n,m)\in\Delta\times\sigma}$ converges to $g(\infty,y)$.
That is, $\\{g^{*}(n)(y_{m})\\}_{(n,m)\in\Delta\times\sigma}$ converges to
$g^{*}(\infty)(y)$, which implies
$\\{f_{n}(y_{m})\\}_{(n,m)\in\Delta\times\sigma}$ converges to $f(y)$. Hence
$\\{f_{n}\\}_{n\in\Delta}$ equi-continuously converges to $f$.

Conversely, let $g^{*}$ be continuous. We have to show that its associated map
$g$ is equi-continuous. Let $\\{x_{n}\\}_{n\in\Delta}$ and
$\\{y_{m}\\}_{m\in\sigma}$ be two convergent nets in $X$ and $Y$ respectively
such that $\\{(x_{n},y_{m})\\}_{(n,m)\in\Delta\times\sigma}$ converges to
$(x,y)$. Since $\\{x_{n}\\}_{n\in\Delta}$ converges to $x$ and $g^{*}$ is
continuous, therefore the image net $\\{g^{*}(x_{n})\\}_{n\in\Delta}$
converges to $g^{*}(x)$. Let us define $g^{*}(x_{n})=f_{x_{n}}$ and
$g^{*}(x)=f_{x}$. Then, we have $\\{f_{x_{n}}\\}_{n\in\Delta}$ converges to
$f_{x}$ in $EC(Y,Z)$. Thus by the given hypothesis,
$\\{f_{x_{n}}\\}_{n\in\Delta}$ equi-continuously converges to $f_{x}$. Then
for the convergent net $\\{y_{m}\\}_{m\in\sigma}$ which converges to $y$, we
have $\\{f_{x_{n}}(y_{m})\\}_{(n,m)\in\Delta\times\sigma}$ converges to
$f_{x}(y)$, that is $\\{g(x_{n},y_{m})\\}_{(n,m)\in\Delta\times\sigma}$
converges to $g(x,y)$. Hence $g$ is equi-continuous. Therefore $\mathfrak{T}$
is admissible. ∎

Below, we mention a lemma without proof which is valid for function spaces of
continuous functions as well as of continuous multifunctions[8]. Here
$\mu\geq\tau$, means $\tau\subseteq\mu$.

######  Lemma 4.9.

Let $\tau$ and $\mu$ be two topologies on $EC(Y,Z)$ and $\mu\geq\tau$. Then
admissibility of $\tau$ implies admissibility of $\mu$. On the other hand, if
$\mu$ is splitting, then $\tau$ is also splitting.

  * Proof.

Easy and left for the readers. ∎

Now we provide examples of admissible and splitting topologies using the
results obtained so far.
In our next pair of theorems, we show that open-entourage topology is
admissible whereas point-transitive-entourage topology is splitting.

######  Theorem 4.10.

Let $(Y,\tau)$ and $(Z,\mathcal{U})$ be a topological space and a uniform
space respectively. Then the open-entourage topology on $EC(Y,Z)$ is
admissible.

  * Proof.

Let $(Y,\tau)$ and $(Z,\mathcal{U})$ be a topological space and a uniform
space respectively. We have to show that the open-entourage topology on
$EC(Y,Z)$ is admissible, that is, for each net $\\{f_{n}\\}_{n\in\Delta}$ in
$EC(Y,Z)$, convergence of $\\{f_{n}\\}_{n\in\Delta}$ to $f$ in $EC(Y,Z)$
implies equi-continuous convergence of $\\{f_{n}\\}_{n\in\Delta}$ to $f$.
Let $\\{y_{m}\\}_{m\in\sigma}$ be any convergent net in $Y$ which converges to
$y$. We have to show that the net
$\\{f_{n}(y_{m})\\}_{(n,m)\in\Delta\times\sigma}$ converges to $f(y)$ in
$(Z,\mathcal{U})$.
Let $U$ be any entourage in $\mathcal{U}$. Then there exists some
$U_{0}\in\mathcal{U}$ such that $U_{0}\circ U_{0}\subset U$. As $f$ is equi-
continuous at $y_{m}$ and $U_{0}\in\mathcal{U}$, therefore there exists an
open neighbourhood $V_{0}\in\tau$ of $y_{m}$ such that $f(V_{0})\subseteq
U_{0}[f(y_{m})]$, which implies $f\in(V_{0},U_{0})_{f(y_{m})}$. Since the net
$\\{f_{n}\\}_{n\in\Delta}$ converges to $f$ in $EC(Y,Z)$ and
$(V_{0},U_{0})_{f(y_{m})}$ is a subbasic open neighbourhood of $f$, therefore
$f_{n}\in(V_{0},U_{0})_{f(y_{m})}$ eventually. We have $f_{n}(V_{0})\subseteq
U_{0}[f(y_{m})]$, whence $f_{n}(y_{m})\in U_{0}[f(y_{m})]$ eventually. Hence
we have $(f_{n}(y_{m}),f(y_{m}))\in U_{0}$ eventually.
Now, consider the net $\\{y_{m}\\}_{m\in\sigma}$ converging to $y$ in $Y$. As
$f\in EC(Y,Z)$, the image net $\\{f(y_{m})\\}_{m\in\sigma}$ converges to
$f(y)$, that is, for $U_{0}^{-1}\in\mathcal{U}$, we have $(f(y),f(y_{m}))\in
U_{0}^{-1}$, which implies $(f(y_{m}),f(y))\in U_{0}$ eventually. Hence
$(f_{n}(y_{m}),f(y_{m}))\circ(f(y_{m}),f(y))\in U_{0}\circ U_{0}\subset U$
eventually. Thus we have $(f_{n}(y_{m}),f(y))\in U$ eventually and therefore
the net $\\{f_{n}(y_{m})\\}_{(n,m)\in\Delta\times\sigma}$ converges to $f(y)$
in $Z$. Therefore by Theorem 4.8, the open-entourage topology on $EC(Y,Z)$ is
admissible. ∎

In the next theorem, we show that the point-transitive-entourage topology on
$EC(Y,Z)$ is splitting.

######  Theorem 4.11.

Let $(Y,\tau)$ and $(Z,\mathcal{U})$ be a topological space and a uniform
space respectively. Then the point-transitive-entourage topology on $EC(Y,Z)$
is splitting.

  * Proof.

Let $(Y,\tau)$ and $(Z,\mathcal{U})$ be a topological space and a uniform
space respectively. We have to show that the point-transitive-entourage
topology on $EC(Y,Z)$ is splitting, that is for each net
$\\{f_{n}\\}_{n\in\Delta}$ in $EC(Y,Z)$, equi-continuous convergence to
$\\{f_{n}\\}_{n\in\Delta}$ to $f$ implies convergence of
$\\{f_{n}\\}_{n\in\Delta}$ to $f$ in $EC(Y,Z)$.
Let $(y,U_{t})_{z}$ be any subbasic open neighbourhood of $f$ in $EC(Y,Z)$.
Then, $f(y)\in U_{t}[z]$, that is, $(f(y),z)\in U_{t}$. Let $\\{y_{m}\\}=y$
for each $m\in\sigma$, be a constant net. Then $\\{y_{m}\\}_{m\in\sigma}$
converges to $y$ in $Y$. Since the net $\\{f_{n}\\}_{n\in\Delta}$ equi-
continuously converges to $f$, the net
$\\{f_{n}(y_{m})\\}_{(n,m)\in\Delta\times\sigma}$ converges to $f(y)$ in $Z$,
that is, net $\\{f_{n}(y)\\}_{n\in\Delta}$ converges to $f(y)$. Then for
$U_{t}\in\mathcal{U}$, we have $U_{t}^{-1}\in\mathcal{U}$, which implies
$(f(y),f_{n}(y))\in U_{t}^{-1}$ eventually. Thus we have $(f_{n}(y),f(y))\in
U_{t}$ eventually. Accordingly, we have $(f_{n}(y),f(y))\circ(f(y),z)\in
U_{t}\circ U_{t}\subset U_{t}$. Therefore $(f_{n}(y),z)\in U_{t}$ eventually
which implies $f_{n}\in(y,U_{t})_{z}$ eventually. Hence net
$\\{f_{n}\\}_{n\in\Delta}$ converges to $f$ in $EC(Y,Z)$. Thus point-
transitive-entourage topology on $EC(Y,Z)$ is splitting. ∎

##  5\. Dual Topology For Equi-Continuous Functions

In this section, we introduce the notion of dual topology for the topologies
on $EC(Y,Z)$. We provide here interesting relationships regarding the
splittingness and admissibility of a topology on equi-continuous functions and
its dual.
For a topological space $(Y,\tau)$ and a uniform space $(Z,\mathcal{U})$, let
$f\in EC(Y,Z)$, $U\in\mathcal{U}$ and $y\in Y$. Then by the definition of
equi-continuity, there exists $V\in\tau$ of $y$ such that $f(V)\subseteq
U[f(y)]$. We denote the open set $V$ obtained this way by $U(f,y)$. Now we
define :

$\mathcal{O}_{Z}(Y)=\\{U(f,y)\,:\,U\in\mathcal{U}\,\,,f\in EC(Y,Z),y\in Y\\}$.

######  Definition 5.1.

Let $(Y,\tau)$ and $(Z,\mathcal{U})$ be a topological space and a uniform
space respectively. Let $EC(Y,Z)$ be the set of all equi-continuous functions
from $Y$ to $Z$. Then for subsets $\mathbb{H}\subseteq\mathcal{O}_{Z}(Y)$,
$\mathcal{H}\subseteq EC(Y,Z)$ and $U\in\mathcal{U}$, we define:
$(\mathbb{H},U)=\\{f\in EC(Y,Z)$ $|$ $U(f,y)\in\mathbb{H}$ for each $y\in Y$
$\\}$
$(\mathcal{H},U)=\\{U(f,y)$ $|$ $f\in\mathcal{H}$, $y\in Y$ $\\}.$

Let $(Y,\tau)$ and $(Z,\mathcal{U})$ be a topological space and a uniform
space respectively and $f\in EC(Y,Z)$, $U\in\mathcal{U}$. Then for each $y\in
Y$, there exists $V\in\tau$ such that $f(V)\subseteq U[f(y)]$. Then
$\mathbb{H}=\\{U(f,y)$ $|$ $y\in Y\\}$ is a subset of $\mathcal{O}_{Z}(Y)$,
such that $f\in(\mathbb{H},U)$. Therefore one can always define the sets of
the form $(\mathbb{H},U)$ and $(\mathcal{H},U)$ which are non empty and well
defined.

######  Definition 5.2.

Let $(Y,\tau)$ and $(Z,\mathcal{U})$ be a topological space and a uniform
space respectively. Let $\mathbb{T}$ be a topology on $\mathcal{O}_{Z}(Y)$.
Then we define:

$\mathcal{S}(\mathbb{T})=\\{(\mathbb{H},U)$ $|$
$\mathbb{H}\in\mathbb{T},U\in\mathcal{U}\\}$.

######  Theorem 5.3.

$\mathcal{S}(\mathbb{T})$ is a subbasis for a topology on $EC(Y,Z)$.

  * Proof.

Let $f\in EC(Y,Z)$. Then for $y\in Y$, $U\in\mathcal{U}$, there exists
$V_{y}\in\tau$ such that $f(V_{y})\subseteq U[f(y)]$. Consider $V_{y}=U(f,y)$.
As $V_{y}\in\mathcal{O}_{Z}(Y)$ and $\mathbb{T}$ is a topology on
$\mathcal{O}_{Z}(Y)$, therefore there exists an open set $\mathbb{H}_{y}$,
such that $V_{y}=U(f,y)\in\mathbb{H}_{y}$. Let
$\mathbb{H}=\displaystyle{\bigcup_{y\in Y}}\mathbb{H}_{y}$. Then
$f\in(\mathbb{H},U)$. Hence $EC(Y,Z)=\bigcup\mathcal{S}(\mathbb{T})$.
Therefore $\mathcal{S}(\mathbb{T})$ is a subbasis for a topology on $EC(Y,Z)$.
∎

Now, we provide a topology on $\mathcal{O}_{Z}(Y)$ using the topology on
$EC(Y,Z)$.

######  Theorem 5.4.

Let $\mathfrak{T}$ be a topology on $EC(Y,Z)$. Then

$\mathcal{S}(\mathfrak{T})=\\{(\mathcal{H},U)$ $|$
$\mathcal{H}\in\mathfrak{T}$, $U\in\mathcal{U}\\}$

is a subbasis for a topology on $\mathcal{O}_{Z}(Y)$.

  * Proof.

Let $U(f,y)\in\mathcal{O}_{Z}(Y)$. Clearly $f\in EC(Y,Z)$ and hence
$f\in\mathcal{H}$ for some $\mathcal{H}\in\mathfrak{T}$. Then
$U(f,y)\in(\mathcal{H},U)$. Therefore
$\mathcal{O}_{Z}(Y)=\bigcup(\mathcal{H},U)$. Hence $\mathcal{S}(\mathfrak{T})$
is a subbasis for a topology on $\mathcal{O}_{Z}(Y)$. ∎

The topologies defined above on $EC(Y,Z)$ and $\mathcal{O}_{Z}(Y)$ are denoted
by $\mathfrak{T}(\mathbb{T})$ and $\mathbb{T}(\mathfrak{T})$ respectively. We
shall refer these topologies as dual to $\mathbb{T}$ and $\mathfrak{T}$
respectively.
Now we define the splittingness and admissibility on $\mathcal{O}_{Z}(Y)$ and
investigate the possible relationships between a topology on $EC(Y,Z)$ and its
dual and vice-versa.

######  Definition 5.5.

Let $(X,\tau)$ and $(Y,\mu)$ be two topological spaces. A multifunction
$F:X\to Y$ is called

  1. $(i)$

upper semi continuous (or $u.s.c.$, in brief) at $x\in X$ if for each open set
$V\subseteq Y$ with $F(x)\subseteq V$, there exists an open set $U$ of $X$
such that $x\in U$ and $F(U)\subseteq V$;

  2. $(ii)$

lower semi continuous (or $l.s.c$, in brief) at $x\in X$ if for each open set
$V\subseteq Y$ with $F(x)\cap V\neq\emptyset$, there exists an open set $U$ of
$X$ such that $x\in U$ and $F(u)\cap V\neq\emptyset$ for every $u\in U$;

  3. $(iii)$

continuous at $x\in X$, if it is both $u.s.c.$ and $l.s.c.$ at $x$;

  4. $(iv)$

continuous (resp. $u.s.c.$, $l.s.c.$) if it is continuous (resp. $u.s.c.$,
$l.s.c.$) at each point of $X$.

######  Definition 5.6.

Let $(Y,\tau)$ and $(Z,\mathcal{U})$ be a topological space and a uniform
space respectively. Let $(X,\mu)$ be another topological space. Let
$g\,:X\times Y\to Z$ and $g^{*}\,:X\to EC(Y,Z)$ be two associated maps. . Then
we define a multifunction
$\overline{g}\,:X\times\mathcal{U}\to\mathcal{O}_{Z}(Y)$ by
$\overline{g}(x,U)=\\{U(g^{*}(x),y)\,|\,y\in Y\\}=\\{U(g_{x},y)\,|\,y\in
Y\\}$, for every $x\in X$ and $U\in\mathcal{U}$.

######  Definition 5.7.

Let $(Y,\tau)$ and $(Z,\mathcal{U})$ be a topological space and a uniform
space respectively. Let $(X,\mu)$ be another topological space. A
multifunction $M\,:X\times\mathcal{U}\to\mathcal{O}_{Z}(Y)$ is called upper
semi continuous with respect to the first variable if the map
$M_{U}\,:X\to\mathcal{O}_{Z}(Y)$ defined by $M_{U}(x)=M(x,U)$ is upper semi
continuous for every $x\in X$ and for a fixed $U\in\mathcal{U}$.

Now, we are in position to define the admissibility and splittingness of the
topological space $(\mathcal{O}_{Z}(Y),\mathbb{T})$.

######  Definition 5.8.

Let $(Y,\tau)$ and $(Z,\mathcal{U})$ be a topological space and a uniform
space respectively. Let $(X,\mu)$ be another topological space. Then topology
$\mathbb{T}$ on $\mathcal{O}_{Z}(Y)$ is called

  1. $(i)$

splitting if equi-continuity of the map $g\,:X\times Y\to Z$ implies upper
semi continuity with respect to the first variable of the map
$\overline{g}\,:X\times\mathcal{U}\to\mathcal{O}_{Z}(Y)$;

  2. $(ii)$

admissible if for every map $g^{*}\,:X\to EC(Y,Z)$, upper semi continuity with
respect to the first variable of the map
$\overline{g}\,:X\times\mathcal{U}\to\mathcal{O}_{Z}(Y)$ implies equi-
continuity of the map $g\,:X\times Y\to Z$.

In the remaining part of this section, we investigate how duality links the
admissibility and splittingness of a topology on $EC(Y,Z)$ and that on
$\mathcal{O}_{Z}(Y)$.

######  Theorem 5.9.

A topology $\mathbb{T}$ on $\mathcal{O}_{Z}(Y)$ is splitting if and only if
its dual topology $\mathfrak{T}(\mathbb{T})$ on $EC(Y,Z)$ is splitting.

  * Proof.

Let $(\mathcal{O}_{Z}(Y),\mathbb{T})$ be splitting, that is, for every
topological space $(X,\mu)$, equi-continuity of the map $g\,:X\times Y\to Z$
implies upper semi continuity with respect to the first variable of the map
$\overline{g}\,:X\times\mathcal{U}\to\mathcal{O}_{Z}(Y)$. We have to show that
the topology $\mathfrak{T}(\mathbb{T})$ on $EC(Y,Z)$ is splitting, that is for
every topological space $X$, equi-continuity of the map $f:X\times Y\to Z$
implies continuity of the associated map $f^{*}:X\to EC(Y,Z)$. Therefore, it
is sufficient to show that upper semi continuity with respect to the first
variable of the map $\overline{g}:X\times\mathcal{U}\to\mathcal{O}_{Z}(Y)$
implies continuity of the associated map $g^{*}:X\to EC(Y,Z)$.
Let $x\in X$ and $(\mathbb{H},U)\in\mathfrak{T}(\mathbb{T})$ be a subbasic
open neighbourhood of $g^{*}(x)$. Then $g^{*}(x)\in(\mathbb{H},U)$, which
implies $U(g^{*}(x),y)\in\mathbb{H}$ for each $y\in Y$. Therefore
$U(g_{x},y)\in\mathbb{H}$ and hence $\overline{g}(x,U)\subseteq\mathbb{H}$.
Now $\overline{g}:X\times\mathcal{U}\to\mathcal{O}_{Z}(Y)$ is upper semi
continuous with respect the first variable and $\mathbb{H}$ is an open
neighbourhood of $\overline{g}_{U}(x)$. Hence there exists an open
neighbourhood $V$ of $x$ such that $\overline{g}_{U}(V)\subseteq\mathbb{H}$.
Now, for an element $x^{\prime}\in V$, we have
$\overline{g}_{U}(x^{\prime})\subseteq\mathbb{H}$. Therefore
$\overline{g}(x^{\prime},U)\subseteq\mathbb{H}$ and hence
$U(g_{x^{\prime}},y)\in\mathbb{H}$ for each $y\in Y$. That is,
$U(g^{*}(x^{\prime}),y)\in\mathbb{H}$ for every $x^{\prime}\in V$, which
implies $g^{*}(x^{\prime})\in(\mathbb{H},U)$ for every $x^{\prime}\in V$. Thus
$g^{*}(V)\subseteq(\mathbb{H},U)$. Therefore the map $g^{*}$ is continuous.
Conversely, let $\mathfrak{T}(\mathbb{T})$ be splitting, we have to show that
the topology $\mathbb{T}$ is splitting. For this, it is sufficient to show
that $\overline{g}:X\times\mathcal{U}\to\mathcal{O}_{Z}(Y)$ is upper semi
continuous with respect to the first variable provided that the map
$g^{*}\,:X\to EC(Y,Z)$ is continuous. Let, for a fixed $U\in\mathcal{U}$ and
$x\in X$, $\mathbb{H}\in\mathcal{O}_{Z}(Y)$ be an open neighbourhood of
$\overline{g}(x,U)$. That is $\overline{g}(x,U)\subseteq\mathbb{H}$ which
implies $U(g_{x},y)\in\mathbb{H}$ for each $y\in Y$. Therefore
$U(g^{*}(x),y)\in\mathbb{H}$ for each $y\in Y$. Thus we have
$g^{*}(x)\in(\mathbb{H},U)$. Now the map $g^{*}$ is given to be continuous and
$(\mathbb{H},U)$ is an open neighbourhood of $g^{*}(x)$. Thus there exists an
open neighbourhood $V$ of $x$ such that $g^{*}(V)\subseteq(\mathbb{H},U)$. Now
for, any $x^{\prime}\in V$, we have $g^{*}(x^{\prime})\in(\mathbb{H},U)$.
Therefore, $U(g^{*}(x^{\prime}),y)=U(g_{x^{\prime}},y)\in\mathbb{H}$ for every
$x^{\prime}\in V$. Hence, we have
$\overline{g}_{U}(x^{\prime})\subseteq\mathbb{H}$, for all $x^{\prime}\in V$.
Hence $\overline{g}_{U}(V)\subseteq\mathbb{H}$. Hence the map $\overline{g}$
is upper semi continuous with respect to the first variable. Thus, the
topology $\mathbb{T}$ is a splitting. ∎

######  Theorem 5.10.

A topology $\mathbb{T}$ on $\mathcal{O}_{Z}(Y)$ is admissible if and only if
its dual topology $\mathfrak{T}(\mathbb{T})$ on $EC(Y,Z)$ is admissible.

  * Proof.

Let the topology $\mathbb{T}$ on $\mathcal{O}_{Z}(Y)$ be admissible, that is,
for every topological space $(X,\mu)$ and for every map $g^{*}:X\to EC(Y,Z)$,
upper semi continuity of the map
$\overline{g}:X\times\mathcal{U}\to\mathcal{O}_{Z}(Y)$ with respect the first
variable implies equi-continuity of the map $g:X\times Y\to Z$. We have to
show that the topology $\mathfrak{T}(\mathbb{T})$ is admissible, that is
continuity of $g^{*}:X\to EC(Y,Z)$ implies equi-continuity of its associated
map $g\,:X\times Y\to Z$. Thus it is sufficient to prove that
$\overline{g}:X\times\mathcal{U}\to\mathcal{O}_{Z}(Y)$ is upper semi
continuous with respect to the first variable provided the map $g^{*}:X\to
EC(Y,Z)$ is continuous.
Let us have, for fixed $U\in\mathcal{U}$ and $x\in X$, a subbasic open
neighbourhood $\mathbb{H}$ of $\overline{g}(x,U)$. Therefore
$\overline{g}(x,U)\subseteq\mathbb{H}$. That is,
$\overline{g}_{U}(x)\subseteq\mathbb{H}$ which implies
$U(g_{x},y)\in\mathbb{H}$ for each $y\in Y$. Thus $g^{*}(x)\in(\mathbb{H},U)$.
Since the map $g^{*}$ is given to be continuous and $(\mathbb{H},U)$ is a
subbsaic open neighbourhood of $g^{*}(x)$, therefore there exists an open
neighbourhood $V$ of $x$ such that $g^{*}(V)\subseteq(\mathbb{H},U)$. Now, for
$x^{\prime}\in V$, we have $g^{*}(x^{\prime})\in(\mathbb{H},U)$, that is
$U(g^{*}(x^{\prime}),y)=U(g_{x^{\prime}},y)\in\mathbb{H}$ for each $y\in Y$.
Thus $\overline{g}_{U}(x^{\prime})\subseteq\mathbb{H}$ for all $x^{\prime}\in
V$. Hence, $\overline{g}_{U}(V)\subseteq\mathbb{H}$. Therefore the map
$\overline{g}$ is upper semi continuous with respect to the first variable.
Hence the topology $\mathfrak{T}(\mathbb{T})$ is admissible.
Conversely, let $\mathfrak{T}(\mathbb{T})$ be admissible , we have to show
that the topology $\mathbb{T}$ on $\mathcal{O}_{Z}(Y)$ is admissible. For
this, it is sufficient to show that upper semi continuity with respect to the
first variable of the map
$\overline{g}:X\times\mathcal{U}\to\mathcal{O}_{Z}(Y)$ implies continuity of
the map $g^{*}:X\to EC(Y,Z)$.
Let $x\in X$ and $(\mathbb{H},U)$ be a subbasic open neighbourhood of
$g^{*}(x)$, that is $g^{*}(x)\in(\mathbb{H},U)$. Thus
$U(g^{*}(x),y)\in\mathbb{H}$ for every $y\in Y$. Hence
$\overline{g}_{U}(x)\subseteq\mathbb{H}$. Now the map $\overline{g}$ is given
to be upper semi continuous with respect to the first variable and
$\mathbb{H}$ is a subbaisc open neighbourhood of $\overline{g}_{U}(x)$. Thus
there exists an open neighbourhood $V$ of $x$ such that
$\overline{g}_{U}(V)\subseteq\mathbb{H}$. Hence for $x^{\prime}\in V$, we have
$\overline{g}_{U}(x^{\prime})\subseteq\mathbb{H}$, which implies
$\overline{g}(x^{\prime},U)\subseteq\mathbb{H}$. Hence
$U(g_{x^{\prime}},y)=U(g^{*}(x^{\prime}),y)\in\mathbb{H}$ for each $y\in Y$.
Therefore $g^{*}(x^{\prime})\in(\mathbb{H},U)$ for all $x^{\prime}\in V$.
Therefore $g^{*}(V)\subseteq(\mathbb{H},U)$. Thus the topology $\mathbb{T}$ is
admissible. ∎

In our next set of theorems, we investigate the relationship between a
topology on $EC(Y,Z)$ and its dual.

######  Theorem 5.11.

A topology $\mathfrak{T}$ on $EC(Y,Z)$ is splitting if and only if its dual
topology $\mathbb{T}(\mathfrak{T})$ is splitting.

  * Proof.

Let $\mathfrak{T}$ be a splitting topology on $EC(Y,Z)$. We have to show that
its dual topology $\mathbb{T}(\mathfrak{T})$ is also splitting. For this, it
is sufficient to prove that continuity of the map $g^{*}:X\to EC(Y,Z)$ implies
upper semi continuity of the map
$\overline{g}:X\times\mathcal{U}\to\mathcal{O}_{Z}(Y)$ with respect to the
first variable.
Let $x\in X$ and $\mathcal{H}\in\mathfrak{T}$ be an open neighbourhood of
$g^{*}(x)$. Then for any fixed $U\in\mathcal{U}$,
$(\mathcal{H},U)\in\mathbb{T}(\mathfrak{T})$ is an open neighbourhood of
$\overline{g}(x,U)$. That is, $\overline{g}(x,U)\subseteq(\mathcal{H},U)$. Now
$\overline{g}(x,U)=\\{U(g_{x},y)\,|\,y\in Y\\}\subseteq(\mathcal{H},U)$, hence
$U(g_{x},y)\in(\mathcal{H},U)$ for each $y\in Y$ by definition. This implies
$g^{*}(x)\in\mathcal{H}$. Since the map $g^{*}$ is given to be continuous and
$\mathcal{H}$ is an open neighbourhood of $g^{*}(x)$, therefore there exists
an open neighbourhood $V$ of $x$ such that $g^{*}(V)\subseteq\mathcal{H}$.
Now, consider an element $x^{\prime}\in V$, we have
$g^{*}(x^{\prime})\in\mathcal{H}$, that is
$U(g_{x^{\prime}},y)\in(\mathcal{H},U)$ for each $y\in Y$. Hence
$\overline{g}(x^{\prime},U)\subseteq(\mathcal{H},U)$, for all $x^{\prime}\in
V$. Therefore $\overline{g}_{U}(V)\subseteq(\mathcal{H},U)$ and the map
$\overline{g}$ is upper semi continuous with respect to the first variable.
Hence the result.
Conversely, let the topology $\mathbb{T}(\mathfrak{T})$ be a splitting
topology. We have to show that the topology $\mathfrak{T}$ on $EC(Y,Z)$ is
splitting. It is equivalent to show that the map $g^{*}:X\to EC(Y,Z)$ is
continuous provided the map
$\overline{g}:X\times\mathcal{U}\to\mathcal{O}_{Z}(Y)$ is upper semi
continuous.
Let $x\in X$ and $\mathcal{H}$ be an open neighbourhood of $g^{*}(x)$, that
is, $g^{*}(x)\in\mathcal{H}$. For any fixed $U\in\mathcal{U}$, we have
$U(g_{x},y)\in(\mathcal{H},U)$ for each $y\in Y$. Therefore
$\overline{g}(x,U)\subseteq(\mathcal{H},U)$ for a fixed $U\in\mathcal{U}$.
Since the map $\overline{g}$ is given to be continuous with respect to the
first variable, there exists an open neighbourgood $V$ of $x$ such that
$\overline{g}_{U}(V)\subseteq(\mathcal{H},U)$. Now, for $x^{\prime}\in V$, we
have $\overline{g}_{U}(x^{\prime})\subseteq(\mathcal{H},U)$ which implies
$U(g_{x^{\prime}},y)\in(\mathcal{H},U)$ for each $y\in Y$. Therefore
$g^{*}(x^{\prime})\in\mathcal{H}$ for every $x^{\prime}\in V$. That is,
$g^{*}(V)\subseteq\mathcal{H}$. Hence the map $g^{*}$ is continuous. ∎

######  Theorem 5.12.

A topology $\mathfrak{T}$ on $EC(Y,Z)$ is admissible if and only if its dual
topology $\mathbb{T}(\mathfrak{T})$ is admissible.

  * Proof.

Left for the reader. ∎

######  Conclusion.

In this paper, we have studied topological structures on the family of equi-
continuous mappings between a topological space and a uniform space. Important
properties such as splittingness, admissibility etc. are introduced for such
spaces and their characterizations are provided using net-theory. We have
shown that similar studies can be carried out for pseudo-dislocated equi-
continuous mappings also. It will be interesting to investigate the existence
of the greatest splitting topology for such spaces. At the same time, the
effect of duality on the existence of the greatest splitting topology needs to
be investigated.

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