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\begin{document}

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\section*{topological base}

\begin{quote}%
This page discusses bases for the topology on [[topological spaces]]. For the concept of topological [[linear basis]] see at \emph{[[basis in functional analysis]]}. For bases on [[sites]], that is for [[Grothendieck topologies]], see at \emph{[[Grothendieck pretopology]]}.


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

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

[[!include topology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{ForTopologicalSpaces}{Definition}\dotfill \pageref*{ForTopologicalSpaces} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{generation}{Generating topologies}\dotfill \pageref*{generation} \linebreak
\noindent\hyperlink{ForSites}{Relation to Grothendieck topologies and coverages}\dotfill \pageref*{ForSites} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

A \emph{base} or \emph{subbase} for a [[topological space]] is a way of generating its topology from something simpler. This is the application to topology of the general concept of [[base]].

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

Let $X$ be a [[topological space]], and let $\tau$ be its collection of [[open subsets]] (its `topology').

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{base} or \textbf{basis} for (or ``of'') $X$ (or $\tau$) is a [[subset|collection]] $B \subset \tau$ -- whose members are called \textbf{basic open subsets} or \textbf{generating open subsets} -- such that every open subset is a [[union]] of basic ones.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
A \textbf{subbase} for (or ``of'') $X$ (or $\tau$) is a subcollection $S \subset \tau$ -- whose members are called \textbf{subbasic open subsets} -- such that every open subset is a [[union]] of [[finitary intersections]] of subbasic ones.

\end{defn}
If only the [[underlying set]] of $X$ is given, then a \textbf{base} or \textbf{subbase} on this [[set]] is any collection of [[subsets]] of $X$ that is a base or subbase for \emph{some} topology on $X$. See \hyperlink{generation}{below} for a characterisation of which collections these can be.

Now fix a [[point]] $a$ in $X$.

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{local base} or \textbf{base of neighborhoods} or \textbf{fundamental system of neighborhoods} for (or ``of'') $X$ (or $\tau$) at $a$ is a subcollection $B \subset \tau$ -- whose members are called \textbf{basic neighborhoods} or \textbf{generating neighborhoods} of $a$ -- such that every basic neighborhood of $a$ is a [[neighborhood]] and every neighborhood of $a$ is a [[superset]] of some basic neighborhood.

\end{defn}
We may also allow basic neighborhoods to be non-open, but this really doesn't make any difference; any local base may be refined to a local base of open neighborhoods, and most local bases in practices already come that way.

A \textbf{local subbase} at $a$ is a family of neighbourhoods $a$ such that each neighbourhood of $a$ contains a finite [[intersection]] of elements of the family.

\begin{defn}
\label{}\hypertarget{}{}
The minimum [[cardinality]] of a base of $X$ is the \textbf{weight} of $X$. The minimum cardinality of a base of neighborhoods at $a$ is the \textbf{character} of $X$ at $a$. The [[supremum]] of the characters at all points of $X$ is the \textbf{character} of $X$.

\end{defn}
We have assumed the [[axiom of choice]] to simplify the description of this concept; but in general one must speak of classes of cardinalities rather than individual cardinalities.

If the character of $X$ is [[countable set|countable]], we say that $X$ satisfies the [[first-countable space|first axiom of countability]]; if the weight is countable, we say that $X$ satisfies the [[second-countable space|second axiom of countability]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{}\hypertarget{}{}
For the [[discrete topology]] on a set $X$, the collection of all [[singleton]] subsets is a base, and the singleton $\{x\}$ is a local base at $x$. Thus every discrete space is first-countable, but only [[countable set|countable]] discrete spaces are second-countable.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
For every [[metric space]], in particular every [[paracompact space|paracompact]] [[Riemannian manifold]], the collection of [[open subsets]] that are [[open balls]] forms a base for the topology. (For instance, a base for the topology on the [[real line]] is given by the collection of open intervals $(a,b) \subset \mathbb{R}$.) Similarly, the collection of open balls containing a given point is a local basis at that point.

\end{example}
\begin{remark}
\label{}\hypertarget{}{}
This means that [[covering]] families consisting of such \emph{basic} open subsets are [[good open covers]].

\end{remark}
\begin{example}
\label{}\hypertarget{}{}
Refining the previous example, every [[metric space]] has a basis consisting of the [[open balls]] with \emph{[[rational number|rational]]} radius. (For instance, a base for the topology on the [[real line]] is given by the collection of open intervals $(a,b) \subset \mathbb{R}$ where $b - a$ is rational.) Similarly, the collection of open balls with rational radius containing a given point is a local base at that point. Therefore, every metric space is first-countable.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
Now consider a [[separable space|separable]] metric space; that is, we have a [[dense subset]] $D$ which is [[countable set|countable]]. Now the space has a basis consisting of the [[open balls]] with [[rational number|rational]] radius and centres in $D$. (For instance, a base for the topology on the [[real line]] is given by the collection of open intervals $(a,b) \subset \mathbb{R}$ where $a$ and $b$ are rational.) Therefore, every separable metric space is second-countable.

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

\hypertarget{generation}{}\subsubsection*{{Generating topologies}}\label{generation}

Let $X$ be simply a [[set]].

\begin{prop}
\label{Recognition}\hypertarget{Recognition}{}
\textbf{(recognition of topological bases)}

A collection $B$ of [[subsets]] of $X$ is a base for \emph{some} topology on $X$ iff these conditions are met:

\begin{itemize}%
\item The elements of $B$ cover $X$;
\item For any $U, V \in B$ and any point $x \in U \cap V$ there is a $W \in B$ such that $W \subseteq U \cap V$ and $x \in W$.

\end{itemize}
\end{prop}
These conditions amount to saying that for each $x\in X$, the subcollection of those $U\in B$ such that $x\in U$ is a base for a [[filter]] on $X$ (which is then the [[neighborhood]] filter of $x$) --- in other words, that these subcollections are ``colaxly closed'' under finite intersections.

\begin{prop}
\label{}\hypertarget{}{}
\emph{Every} collection $S$ of subsets of $X$ is a subbase for some topology on $X$.

\end{prop}
A subbase naturally generates a base (for the same topology) by [[Moore closure|closing]] it under finitary intersections. (The resulting base will actually be closed under intersection.)

\hypertarget{ForSites}{}\subsubsection*{{Relation to Grothendieck topologies and coverages}}\label{ForSites}

If one thinks of the topology on $X$ as being encoded in the standard [[Grothendieck topology]] that it induces on its [[category of open subsets]] $Op(X)$, then a base for the topology induces a \emph{[[coverage]]} on $Op(X)$, whose covering families are the open covers by basic open subsets, which generates this Grothendieck topology.

This coverage is \emph{not} in general a [[basis for the Grothendieck topology]], because a base for a topological space is in general not closed under [[intersection]] with arbitrary [[open subsets]]; a coverage is only a basis if is stable under [[pullback]] (here, closed under these intersections) and transitive. Unfortunately the established terminology ``basis'' in [[topology]] and [[topos theory]] is not quite consistent with the inclusion of topological spaces into topos theory: ``basis'' in topology corresponds to ``coverage'' in topos theory, not to ``basis'' in topos theory.

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

\begin{itemize}%
\item [[neighbourhood base]]

\end{itemize}
[[!redirects base for a topology]] [[!redirects basis for a topology]] [[!redirects bases for a topology]] [[!redirects bases for topologies]] [[!redirects base of a topology]] [[!redirects basis of a topology]] [[!redirects bases of a topology]] [[!redirects bases of topologies]]

[[!redirects topological basis]] [[!redirects topological basises]]

[[!redirects base for a topological space]] [[!redirects basis for a topological space]] [[!redirects bases for a topological space]] [[!redirects bases for topological spaces]] [[!redirects base of a topological space]] [[!redirects basis of a topological space]] [[!redirects bases of a topological space]] [[!redirects bases of topological spaces]] [[!redirects topological base]] [[!redirects topological bases]] [[!redirects base for the topology]] [[!redirects basis for the topology]] [[!redirects bases for the topology]] [[!redirects base of the topology]] [[!redirects basis of the topology]] [[!redirects bases of the topology]] [[!redirects base for its topology]] [[!redirects basis for its topology]] [[!redirects bases for its topology]] [[!redirects base of its topology]] [[!redirects basis of its topology]] [[!redirects bases of its topology]]

[[!redirects subbase for a topology]] [[!redirects subbasis for a topology]]

[[!redirects sub-basis for a topology]] [[!redirects sub-bases for a topology]]

[[!redirects sub-base of a topology]] [[!redirects sub-bases of a topology]]

[[!redirects subbases for a topology]] [[!redirects subbases for topologies]] [[!redirects subbase of a topology]] [[!redirects subbasis of a topology]] [[!redirects subbases of a topology]] [[!redirects subbases of topologies]] [[!redirects subbase for a topological space]] [[!redirects subbasis for a topological space]] [[!redirects subbases for a topological space]] [[!redirects subbases for topological spaces]] [[!redirects subbase of a topological space]] [[!redirects subbasis of a topological space]] [[!redirects subbases of a topological space]] [[!redirects subbases of topological spaces]] [[!redirects topological subbase]] [[!redirects topological subbases]] [[!redirects subbase for the topology]] [[!redirects subbasis for the topology]] [[!redirects subbases for the topology]]

[[!redirects local base]] [[!redirects local basis]] [[!redirects local bases]] [[!redirects base of neighborhoods]] [[!redirects basis of neighborhoods]] [[!redirects bases of neighborhoods]] [[!redirects base of neighbourhoods]] [[!redirects basis of neighbourhoods]] [[!redirects bases of neighbourhoods]] [[!redirects fundamental system of neighborhoods]] [[!redirects fundamental systems of neighborhoods]] [[!redirects fundamental system of neighbourhoods]] [[!redirects fundamental systems of neighbourhoods]]

[[!redirects sub-base for a topology]] [[!redirects sub-bases for a topology]]



\end{document}