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\section*{complete space}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{complete_spaces}{}\section*{{Complete spaces}}\label{complete_spaces}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{completion}{Completion}\dotfill \pageref*{completion} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_compact_spaces}{Relation to compact spaces}\dotfill \pageref*{relation_to_compact_spaces} \linebreak
\noindent\hyperlink{relation_to_baire_spaces}{Relation to Baire spaces}\dotfill \pageref*{relation_to_baire_spaces} \linebreak
\noindent\hyperlink{generalization}{Generalization}\dotfill \pageref*{generalization} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A [[space]] (with ``space'' taken in a sense relevant to the field of [[topology]]) is \emph{complete} (or \emph{Cauchy-complete}) if every [[sequence]], [[net]], or [[filter]] that should converge really does [[convergence|converge]]. We identify the sequences, nets, or filters that ``should'' converge as the \emph{[[Cauchy sequence|Cauchy]]} ones.

A space that is not complete has ``gaps'' that may be filled to form its \emph{completion}; it is rather natural to make the space (or equivalently its underlying topological space) [[Hausdorff space|Hausdorff]] at the same time. Forming the completion of a Hausdorff space is an important example of [[completion]] in the general abstract sense.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

A space (which may be a [[metric space]], a [[Cauchy space]], or anything in between) is \textbf{Cauchy-complete}, or simply \textbf{complete}, if every [[Cauchy filter]] converges, equivalently if every [[Cauchy net]] converges. A space is \textbf{sequentially complete} if every [[Cauchy sequence]] converges. Note that a sequentially complete metric space must be complete (in [[classical mathematics]]), but this does not hold for more general spaces (nor even for metric spaces in [[constructive mathematics]]).

A space is \textbf{topologically complete} if its [[underlying topological space]] is completely [[metrizable space|metrizable]]. There are various other notions related to this. See [[topologically complete space]].

\hypertarget{completion}{}\subsection*{{Completion}}\label{completion}

The set $\mathcal{C}X$ of Cauchy filters on a space $X$ may generally be given the same sort of structure as $X$ itself has, and this space will be complete. Exactly how to do this depends on what structure $X$ is supposed to have, of course, and one can make the general statement false by requiring something artificial as the structure in question, most extremely the structure of being a specific non-complete space. But it works for most natural categories of spaces.

The general idea is this: every point in $X$ generates a principal [[ultrafilter]] (consisting of those sets to which the point belongs), so there is a natural map from $X$ to $\mathcal{C}X$. Furthermore, this map is a morphism of the appropriate structure, which in particular makes it [[Cauchy-continuous map|Cauchy-continuous]] (preserving Cauchy filters) and continuous (preserving limits). So all of the limits in $X$ still exist in $\mathcal{C}X$, but now each Cauchy filter in $X$ (having become both a Cauchy filter in $\mathcal{C}$ and a point in $\mathcal{C}$) has a limit as well. The additional Cauchy filters based on the additional points in $\mathcal{C}X$ will also have a limit in $\mathcal{C}X$, essentially because $\mathcal{C}$ is a [[monad]] (so a Cauchy filter of Cauchy filters folds into a single Cauchy filter).

There is a problem that $\mathcal{C}X$ is rather larger than necessary; for example, all of the filters that converge to a given point in $X$ (not just the free ultrafilter at that point) exist in $\mathcal{C}X$ and converge to one another. But you can take a [[quotient space|quotient]] of $\mathcal{C}X$ to make it Hausdorff, obtaining the \textbf{Hausdorff completion} of $X$. In case $X$ was not Hausdorff to begin with, one can sometimes also force the quotient to leave in just as much redundancy as $X$ has but no more, obtain a straight \textbf{completion} of $X$. But really, it's most natural to make the space Hausdorff at the same time.

Details to come, if I get around to it.

We have a picture like this, where $X$ is the original space, $\mathcal{H}$ gives a Hausdorff quotient, and an overline indicates completion:

\begin{displaymath}
\array {
  &                 & \overline{X} \\
  & \hookrightarrow &              & &#8608; \\
X &                 &              &                 & \mathcal{H}\overline{X} \cong \overline{\mathcal{H}X} \\
  & &#8608;        &              & \hookrightarrow \\
  &                 & \mathcal{H}X
}
\end{displaymath}
(Here the arrows are drawn horizontally to put styles on them; they should all be diagonal in the only possible way.)

At least if $X$ is a [[metric space]], then we can also construct its completion as a [[locale]], the [[localic completion]], whose spatial part is the above space, but which in [[constructive mathematics]] may not be spatial. This is useful to have even if $X$ is already complete.

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

\hypertarget{relation_to_compact_spaces}{}\subsubsection*{{Relation to compact spaces}}\label{relation_to_compact_spaces}

A [[compact space]] is necessarily complete. A space is called \textbf{precompact} if its completion is compact. For metric spaces (or even [[uniform spaces]]), there is a natural notion of a [[totally bounded space]]; in [[classical mathematics]], we have the theorem that a space is totally bounded if and only if it is precompact. Similarly, a space is compact if and only if it is both complete and totally bounded (or in [[constructive mathematics]], both complete and precompact). Thus the purely topological property of compactness is the conjunction of the nontopological properties of completeness and total boundedness.

In some [[constructive analysis|constructive approaches to analysis]] (including most of Brouwer's school and some of Bishop's school), `complete and totally bounded' is taken as the \emph{definition} of `compact', because it holds of examples such as the [[unit interval]] that fail to be compact (in the usual sense) without the [[fan theorem]]. However, in this case, compactness is no longer a [[topological property]]; see [[Bishop-compact space]] for more information. This is reconciled somewhat with the theory of [[localic completion]], in which a uniform space is totally bounded if and only if its localic completion is compact (in the usual sense).

\hypertarget{relation_to_baire_spaces}{}\subsubsection*{{Relation to Baire spaces}}\label{relation_to_baire_spaces}

Every complete metric space is a [[Baire space]]. Since being a Baire space is a [[topological property]], it follows that every [[topologically complete space]] is a Baire space.

Are there (necessarily nonmetrizable) complete uniform spaces that are not Baire spaces?

There is also a topological property of [[Cech-complete space|Čech-completeness]] that is related to this; in particular, a metric space is ech-complete if and only if it is complete, and every ech-complete space is a Baire space. In general, we have these proper implications: topologically complete $\rightarrow$ ech-complete $\rightarrow$ Baire.

\hypertarget{generalization}{}\subsection*{{Generalization}}\label{generalization}

When [[Bill Lawvere]] interpreted (in \hyperlink{Lawvere1973}{Lawvere 1973}) [[metric spaces]] as certain [[enriched categories]], he found that a metric space was complete if and only if every [[adjunction]] of [[bimodules]] over the enriched category is induced by an [[enriched functor]]. Accordingly, this becomes the notion of [[Cauchy-complete category]]. (Note that one \emph{must} say `Cauchy' here, since this is \emph{weaker} than being a [[complete category]], which is based on an incompatible analogy.)

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item a complete [[normed vector space]] is a \emph{[[Banach space]]}

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

\begin{itemize}%
\item A.V. Arkhangel$\prime$skii (1977). \emph{Complete space}. Matematicheskaya entsiklopediya. \href{https://www.encyclopediaofmath.org/index.php/Complete_space}{Updated English version}.


\item [[Bill Lawvere]] (1973). \emph{Metric spaces, generalized logic and closed categories}. Reprinted in [[TAC]], 1986. \href{http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html}{Web}.



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

[[!redirects Cauchy complete space]] [[!redirects Cauchy complete spaces]] [[!redirects Cauchy-complete space]] [[!redirects Cauchy-complete spaces]]

[[!redirects completion of a space]] [[!redirects completion of spaces]] [[!redirects completions of spaces]] [[!redirects Cauchy completion]] [[!redirects Cauchy completions]] [[!redirects Cauchy-completion]] [[!redirects Cauchy-completions]]

[[!redirects complete metric space]] [[!redirects complete metric spaces]] [[!redirects Cauchy complete metric space]] [[!redirects Cauchy complete metric spaces]] [[!redirects Cauchy-complete metric space]] [[!redirects Cauchy-complete metric spaces]] [[!redirects completion of a metric space]] [[!redirects completion of metric spaces]] [[!redirects completions of metric spaces]] [[!redirects Cauchy completion of a metric space]] [[!redirects Cauchy completion of metric spaces]] [[!redirects Cauchy completions of metric spaces]] [[!redirects Cauchy-completion of a metric space]] [[!redirects Cauchy-completion of metric spaces]] [[!redirects Cauchy-completions of metric spaces]] [[!redirects complete metric]] [[!redirects complete metrics]] [[!redirects Cauchy complete metric]] [[!redirects Cauchy complete metrics]] [[!redirects Cauchy-complete metric]] [[!redirects Cauchy-complete metrics]] [[!redirects completion of a metric]] [[!redirects completion of metrics]] [[!redirects completions of metrics]] [[!redirects Cauchy completion of a metric]] [[!redirects Cauchy completion of metrics]] [[!redirects Cauchy completions of metrics]] [[!redirects Cauchy-completion of a metric]] [[!redirects Cauchy-completion of metrics]] [[!redirects Cauchy-completions of metrics]] [[!redirects metric completion]] [[!redirects metric completions]]

[[!redirects complete uniform space]] [[!redirects complete uniform spaces]] [[!redirects Cauchy complete uniform space]] [[!redirects Cauchy complete uniform spaces]] [[!redirects Cauchy-complete uniform space]] [[!redirects Cauchy-complete uniform spaces]] [[!redirects completion of a uniform space]] [[!redirects completion of uniform spaces]] [[!redirects completions of uniform spaces]] [[!redirects Cauchy completion of a uniform space]] [[!redirects Cauchy completion of uniform spaces]] [[!redirects Cauchy completions of uniform spaces]] [[!redirects Cauchy-completion of a uniform space]] [[!redirects Cauchy-completion of uniform spaces]] [[!redirects Cauchy-completions of uniform spaces]] [[!redirects uniform completion]] [[!redirects uniform completions]]

[[!redirects complete Cauchy space]] [[!redirects complete Cauchy spaces]] [[!redirects Cauchy complete Cauchy space]] [[!redirects Cauchy complete Cauchy spaces]] [[!redirects Cauchy-complete Cauchy space]] [[!redirects Cauchy-complete Cauchy spaces]] [[!redirects completion of a Cauchy space]] [[!redirects completion of Cauchy spaces]] [[!redirects completions of Cauchy spaces]] [[!redirects Cauchy completion of a Cauchy space]] [[!redirects Cauchy completion of Cauchy spaces]] [[!redirects Cauchy completions of Cauchy spaces]] [[!redirects Cauchy-completion of a Cauchy space]] [[!redirects Cauchy-completion of Cauchy spaces]] [[!redirects Cauchy-completions of Cauchy spaces]]



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