\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{subsequential space}

\hypertarget{subsequential_spaces}{}\section*{{Subsequential spaces}}\label{subsequential_spaces}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{subsequential space} is a set equipped with a notion of \emph{sequential convergence}, giving it a ``topology'' in an informal sense.

Any [[topological space]] (or more generally, any [[pseudotopological space]]) becomes a subsequential space with its standard notion of convergence, but only for a [[sequential space]] can the topology be recovered from sequential convergence. In the other direction, not every subsequential space is induced by a topological one. Despite these apparent drawbacks, subsequential spaces have a number of advantages; see below.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{subsequential space} is a set $X$ equipped with a relation between sequences and points, called ``converges to,'' with the following properties.

\begin{enumerate}%
\item For every $x\in X$, the constant sequence $(x)$ converges to $x$.


\item If a sequence $(x_n)$ converges to $x$, then so does any subsequence of $x$.


\item If, for some sequence $(x_n)$ and some point $x$, every subsequence of $(x_n)$ contains a further subsequence converging to $x$, then $(x_n)$ itself converges to $x$.



\end{enumerate}
The final property can be stated less [[constructive mathematics|constructively]] as ``if $(x_n)$ does not converge to $x$, then there is a subsequence $(x_{n_k})$ of $(x_n)$ such that no subsequence of $(x_{n_k})$ converges to $x$.''

Note that this definition matches the definition of [[pseudotopological space]] except for the restriction to sequences instead of general [[net]]s. Accordingly, one may call a subsequential space a \textbf{sequential pseudotopological space}.

Subsequential spaces are also known as Kuratowski \emph{limit spaces}, or \emph{L-spaces}; see \hyperlink{Menni}{Menni}.

A subsequential space is said to be \textbf{sequentially [[Hausdorff space|Hausdorff]]} if each sequence converges to at most one limit.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

The definition of a subsequential space is arguably easier and more intuitive than that of a [[topological space]]. Continuity of functions between subsequential spaces is likewise easy to define by preservation of convergent sequences.

As mentioned above, the category $SeqTop$ of [[sequential space|sequential (topological) spaces]] is a full reflective subcategory of the category $SeqPsTop$ of subsequential spaces. Thus, subsequential spaces include many spaces of interest to topologists, including all [[metric space|metrizable]] spaces and all [[CW complex|CW complexes]], and so they can be regarded as a sort of [[nice topological space]].

Not every subsequential space is a sequential (topological) space, but somewhat surprisingly, every sequentially Hausdorff subsequential space \emph{is} necessarily a sequential space. Note, though, that while any Hausdorff space is sequentially Hausdorff, the converse is not true even for sequential spaces (though it is true for first-countable spaces). Also of note is that $SeqTop$ is coreflective in $Top$.

Furthermore, $SeqPsTop$ is \emph{also} a [[nice category of spaces]]: it is [[locally cartesian closed category|locally cartesian closed]] and in fact a [[quasitopos]]. Since it is a ``Grothendieck quasitopos'' (the category of presheaves on a category which are [[sheaf|sheaves]] for one [[Grothendieck topology]] and separated for another one), it is also [[locally presentable category|locally presentable]]. In particular, it is [[complete category|complete]] and cocomplete, and has a small [[generator|generating set]].

Of course, the embedding of $SeqTop$ in $SeqPsTop$ preserves all [[limit]]s, since it has a left adjoint, but somewhat surprisingly it also preserves many [[colimit]]s. In particular, it preserves all the colimits used in the construction of a CW complex; thus it makes no difference whether you carry out the construction of a CW complex in $Top$ and then regard the result as a subsequential space, or carry out the construction in $SeqPsTop$ to begin with.

It follows that the [[geometric realization]] functor from [[simplicial set]]s can equally well be regarded as landing in $Top$, $SeqTop$, or $SeqPsTop$. Of course, it has a singular complex functor as a right adjoint in any of these three cases. In the cases of $SeqTop$ and $SeqPsTop$, geometric realization actually preserves all finite limits; in fact it and the singular complex functor form a [[geometric morphism]] between $SimpSet$ and a Grothendieck topos that contains $SeqPsTop$ as a reflective subcategory (the ``[[topological topos]]'' of Johnstone's paper). Recall that geometric realization landing in $Top$ doesn't even preserve finite products, unless we replace $Top$ by (for instance) compactly generated spaces.

These properties of subsequential spaces should be compared with analogous ones for [[convergence space]]s and their relatives, such as [[pseudotopological space]]s. The category $Conv$ of convergence spaces is also a complete and cocomplete quasitopos (hence, in particular, locally cartesian closed) and includes \emph{all} of $Top$ as a reflective subcategory. However, $Conv$ is not locally presentable and has no generator, and while the embedding of $Top$ into $Conv$ also preserves all limits (since it has a left adjoint), it actually preserves \emph{fewer} colimits than the embedding of $SeqTop$ into $SeqPsTop$. In particular, it does \emph{not} preserve the colimits used in the construction of CW complexes: if you carry out the construction of a CW complex in $Conv$, in general the result won't even be a topological space.

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Peter Johnstone]], \emph{On a topological topos}. Proc. London Math. Soc. (3) 38 (1979) 237--271 doi:\href{https://doi.org/10.1112/plms/s3-38.2.237}{10.1112/plms/s3-38.2.237}


\item [[Matias Menni]] and [[Alex Simpson]], \emph{Topological and Limit-space Subcategories of Countably-based Equilogical Spaces}, Math. Struct. in Comp. Science \textbf{12} (2002) pp739-770, (\href{http://homepages.inf.ed.ac.uk/als/Research/Sources/subcats.pdf}{PDF}), doi:\href{https://doi.org/10.1017/S0960129502003699}{10.1017/S0960129502003699}


\item Sean Moss, \href{https://golem.ph.utexas.edu/category/2014/04/on_a_topological_topos.html}{Blog post} at the $n$-Category café



\end{itemize}
[[!redirects subsequential space]] [[!redirects subsequential spaces]]

[[!redirects sequential pseudotopological space]] [[!redirects sequential pseudotopological spaces]] [[!redirects sequential convergence space]] [[!redirects sequential convergence spaces]]

[[!redirects Kuratowski limit space]] [[!redirects Kuratowski limit spaces]] [[!redirects limit space]] [[!redirects limit spaces]] [[!redirects L-space]] [[!redirects L-spaces]]



\end{document}