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\begin{document}

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\section*{topological manifold}

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

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

[[!include topology - contents]]

\hypertarget{manifolds_and_cobordisms}{}\paragraph*{{Manifolds and cobordisms}}\label{manifolds_and_cobordisms}

[[!include manifolds and cobordisms - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{LocallyEuclideanTopologicalSpace}{Locally Euclidean topological spaces}\dotfill \pageref*{LocallyEuclideanTopologicalSpace} \linebreak
\noindent\hyperlink{topological_manifold}{Topological manifold}\dotfill \pageref*{topological_manifold} \linebreak
\noindent\hyperlink{differentiable_manifolds}{Differentiable manifolds}\dotfill \pageref*{differentiable_manifolds} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{topological manifold} is a [[topological space]] (usually required to be [[Hausdorff topological space|Hausdorff]] and [[paracompact topological space|paracompact]]) which is \emph{locally} [[homeomorphism|homeomorphic]] to a [[Euclidean space]] $\mathbb{R}^n$ equipped with its [[metric topology]].

Often one is interested in extra structure on topological manifolds, that make them for instance into [[differentiable manifolds]] or [[smooth manifolds]] or [[analytic manifolds]] or [[complex manifolds]], etc. See at \emph{[[manifold]]} for more on the general concept.

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

\hypertarget{LocallyEuclideanTopologicalSpace}{}\subsubsection*{{Locally Euclidean topological spaces}}\label{LocallyEuclideanTopologicalSpace}

\begin{defn}
\label{LocallyEuclideanSpace}\hypertarget{LocallyEuclideanSpace}{}
\textbf{([[locally Euclidean topological space]])}

A [[topological space]] $X$ is \emph{[[locally Euclidean topological space|locally Euclidean]]} if every point $x \in X$ has an [[open neighbourhood]] $U_x \supset \{x\}$ which is [[homeomorphism|homeomorphic]] to the [[Euclidean space]] $\mathbb{R}^n$ with its [[metric topology]]:

\begin{displaymath}
\mathbb{R}^n
    \overset{\phantom{AA} \simeq \phantom{AA}}{\longrightarrow}
  U_x
    \subset
  X
  \,.
\end{displaymath}
\end{defn}
The ``local'' [[topological properties]] of Euclidean space are inherited by locally Euclidean spaces:

\begin{prop}
\label{LocalPropertiesOfLocallyEuclideanSpace}\hypertarget{LocalPropertiesOfLocallyEuclideanSpace}{}
\textbf{([[locally Euclidean spaces]] are $T_1$-[[separation axiom|separated]], [[sober topological space|sober]], [[locally connected topological space|locally connected]], [[locally compact topological space|locally compact]])}

Let $X$ be a [[locally Euclidean space]] (def. \ref{LocallyEuclideanSpace}). Then

\begin{enumerate}%
\item $X$ satisfies the $T_1$ [[separation axiom]];


\item $X$ is [[sober topological space|sober]];


\item $X$ is [[locally connected topological space|locally connected]];


\item $X$ is [[locally compact topological space|locally compact]] in the sense that every open neighbourhood of a point contains a [[compact topological space|compact]] neighbourhood.



\end{enumerate}
\end{prop}
\begin{proof}
Regarding the first statement:

Let $x \neq y$ be two distinct points in the locally Euclidean space. We need to show that there is an open neighbourhood $U_x$ around $x$ that does not contain $y$.

By definition, there is a Euclidean open neighbourhood $\mathbb{R}^n \underoverset{\simeq}{\phi}{\to} U_x \subset X$ around $x$. If $U_x$ does not contain $y$, then it already is an open neighbourhood as required. If $U_x$ does contain $y$, then $\phi^{-1}(x) \neq \phi^{-1}(y)$ are equivalently two distinct points in $\mathbb{R}^n$. But Euclidean space, as every [[metric space]], is $T_1$, and hence we may find an open neighbourhood $V_{\phi^{-1}(x)} \subset \mathbb{R}^n$ not containing $\phi^{-1}(y)$. By the nature of the [[subspace topology]], $\phi(V_{\phi^{-1}(x)}) \subset X$ is an open neighbourhood as required.

Regarding the second statement:

We need to show that the map

\begin{displaymath}
Cl(\{-\}) \;\colon\; X \to IrrClSub(X)
\end{displaymath}
that sends points to the [[topological closure]] of their singleton sets is a [[bijection]] with the set of [[irreducible closed subsets]]. By the first statement above the map is [[injective function|injective]] (via \href{separation+axioms#T1InTermsOfClosureOfPoints}{this lemma}).

Hence it remains to see that every irreducible closed subset is the topological closure of a singleton. We will show something stronger: every irreducible closed subset is a singleton.

Let $P \subset X$ be an open proper subset such that if there are two open subsets $U_1, U_2 \subset X$ with $U_1 \cap U_2 \subset P$ then $U_1 \subset P$ or $U_2 \subset P$. By \href{irreducible+closed+subspace#OpenSubsetVersionOfClosedIrreducible}{this prop.} we need to show that there exists a point $x \in X$ such that $P = X \setminus \{x\}$ it its [[complement]].

Now since $P \subset X$ is a proper subset, and since the locally Euclidean space $X$ is covered by Euclidean neighbourhoods, there exists a Euclidean neighbourhood $\mathbb{R}^n \underoverset{\simeq}{\phi}{\to} U \subset X$ such that $P \cap U \subset U$ is a proper subset. In fact this still satisfies the condition that for $U_1, U_2 \underset{\text{open}}{\subset} U$ then $U_1 \cap U_2 \subset P \cap U$ implies $U_1 \subset P \cap U$ or $U_2 \subset P \cap U$. Accordingly, by \href{irreducible+closed+subspace#OpenSubsetVersionOfClosedIrreducible}{that prop.} it follows that $\mathbb{R}^n \setminus \phi^{-1}(P \cap U)$ is an irreducible closed subset of [[Euclidean space]]. Sine [[metric spaces]] are [[sober topological space]] as well as $T_1$-[[separation axiom|separated]], this means that there exists $x \in \mathbb{R}^n$ such that $\phi^{-1}(P \cap U) = \mathbb{R}^n \setminus \{x\}$.

In conclusion this means that the restriction of an irreducible closed subset in $X$ to any Euclidean chart is either empty or a singleton set. This means that the irreducible closed subset must be a disjoint union of singletons that are separated by Euclidean neighbourhoods. But by irreducibiliy, this union has to consist of just one point.

Regarding the third statement:

Let $x \in X$ be a point and $U_x \supset \{x\}$ a neighbourhood. We need to find a [[connected topological space|connected]] open neighbourhood $Cn_x \subset U_x$.

By local Euclideanness, there is also a Euclidean neighboruhood $\mathbb{R}^n \underoverset{\simeq}{\phi}{\to} V_x \subset X$. Since $\phi$ is a [[homeomorphism]], and since $U_x \cap V_x$ is open, also $\phi^{-1}(U_x \cap V_x) \subset \mathbb{R}^n$ is open. This means that there exists an [[open ball]] $B_{\phi^{-1}(x)}^\circ(\epsilon) \subset \phi^{-1}(U_x \cap V_x)$. This is open and connected, and hence so is its homeomorphic image $\phi(B^\circ_{\phi^{-1}(x)}(\epsilon)) \subset X$. This is a connected open neighbourhood of $x$ as required.

Regarding the fourth statement:

Let $x \in X$ be a point and let $U_x \supset \{x\}$ be an open neighbourhood. We need to find a compact neighbourhood $K_x \subset U_x$.

By assumption there exists a Euclidean open neighbourhood $\mathbb{R}^n \underoverset{\simeq}{\phi}{\to} V_x \subset X$. By definition of the [[subspace topology]] the intersection $U_x \cap V_x$ is still open as a subspace of $V_x$ and hence $\phi^{-1}(U_x \cap V_x)$ is an open neighbourhood of $\phi^{-1}(x)  \in \mathbb{R}^n$.

Since Euclidean spaces are locally compact, there exists a compact neighbourhood $K_{\phi^{-1}(x)} \subset \mathbb{R}^n$ (for instance a sufficiently small [[closed ball]] around $x$, which is compact by the [[Heine-Borel theorem]]). Now since [[continuous images of compact spaces are compact]], it follows that also $\phi(K) \subset X$ is a compact neighbourhood.

\end{proof}
But the ``global'' topological properties of Euclidean space are not generally inherited by locally Euclidean spaces. This sounds obvious, but notice that also Hausdorff-ness is a ``global property'':

\begin{remark}
\label{}\hypertarget{}{}
\textbf{(locally Euclidean spaces are not necessarily $T_2$)}

It might superficially seem that every locally Euclidean space (def. \ref{LocallyEuclideanSpace}) is necessarily a [[Hausdorff topological space]], since [[Euclidean space]], like any [[metric space]], is Hausdorff, and since by definition the neighbourhood of every point in a locally Euclidean spaces looks like Euclidean space.

But this is not so, Hausdorffness is a ``non-local condition''.

\end{remark}
\begin{example}
\label{NonHausdorffManifolds}\hypertarget{NonHausdorffManifolds}{}
\textbf{([[non-Hausdorff locally Euclidean spaces]])}

An example of a [[locally Euclidean space]] (def. \ref{LocallyEuclideanSpace}) which is a [[non-Hausdorff topological space]], is the [[line with two origins]].

\end{example}
\begin{lemma}
\label{PathConnectedFromConnectedLocallyEuclideanSpace}\hypertarget{PathConnectedFromConnectedLocallyEuclideanSpace}{}
\textbf{(connected locally Euclidean spaces are path-connected)}

A locally Euclidean space which is [[connected topological space|connected]] is also [[path-connected topological space|path-connected]].

\end{lemma}
\begin{proof}
Fix any $x \in X$. Write $PConn_x(X) \subset X$ for the subset of all those points of $x$ which are connected to $x$ by a path, hence

\begin{displaymath}
PConn_x(X)
  \;\colon\;
  \left\{
    y \in X
   \;\vert\;
   \underset{[0,1] \underoverset{cts}{\gamma}{\to} X }{\exists}
   \left(
      \left(\gamma(0) = x\right)
      \phantom{A} \text{and} \phantom{a}
      \left(
        \gamma(1) = y
      \right)
   \right)
  \right\}
  \,.
\end{displaymath}
Observe now that both $PConn_x(X) \subset X$ as well as its [[complement]] are [[open subsets]]:

To see this it is sufficient to find for every point $y \on PConn_x(X)$ an [[open neighbourhood]] $U_y \supset \{y\}$ such that $U_y \subset PConn_x(X)$, and similarly for the complement.

Now by assumption every point $y \in X$ has a Euclidean neighbourhood $\mathbb{R}^n \overset{\simeq}{\to} U_y \subset X$. Since Euclidean space is path connected, there is for every $z \in U_y$ a path $\tilde \gamma \colon [0,1] \to X$ connecting $y$ with $z$, i.e. with $\tilde \gamma(0) = y$ and $\tilde \gamma(1) = z$. Accordingly the composite path

\begin{displaymath}
\itexarray{
     [0,1] &\overset{\tilde \gamma\cdot\gamma}{\longrightarrow}& X
     \\
     t &\overset{\phantom{AAA}}{\mapsto}&
    \left\{
      \itexarray{
         \gamma(2t) &\vert& t \leq 1/2
         \\
         \tilde(2t-1/2) &\vert& t \geq 1/2
      }
    \right.
  }
\end{displaymath}
connects $x$ with $z \in U_y$. Hence $U_y \subset PConn_x(X)$.

Similarly, if $y$ is not connected to $x$ by a path, then also all point in $U_y$ cannot be connected to $x$ by a path, for if they were, then the analogous concatenation of paths would give a path from $x$ to $y$, contrary to the assumption.

It follows that

\begin{displaymath}
X = PConn_x(C) \sqcup (X \setminus PConn_x(X))
\end{displaymath}
is a decomposition of $X$ as the [[disjoint union]] of two open subsets. By the assumption that $X$ is connected, exactly one of these open subsets is empty. Since $PConn_x(X)$ is not empty, as it contains $x$, it follows that its compement is empty, hence that $PConn_x(X) = X$, hence that $(X,\tau)$ is path connected.

\end{proof}
\begin{prop}
\label{RegularityConditionsForTopologicalManifoldsComparison}\hypertarget{RegularityConditionsForTopologicalManifoldsComparison}{}
\textbf{(equivalence of regularity conditions for Hausdorff locally Euclidean spaces)}

Let $X$ be a [[locally Euclidean space]] (def. \ref{LocallyEuclideanSpace}) which is [[Hausdorff topological space|Hausdorff]].

Then the following are equivalent:

\begin{enumerate}%
\item $X$ is [[sigma-compact topological space|sigma-compact]],


\item $X$ is [[second-countable topological space|second-countable]],


\item $X$ is [[paracompact topological space|paracompact]] and has a [[countable set]] of [[connected components]],



\end{enumerate}
\end{prop}
\begin{proof}
Generally, observe that $X$ is [[locally compact]]: By prop. \ref{LocalPropertiesOfLocallyEuclideanSpace} every locally Euclidean space is locally compact in the sense that every point has a [[neighbourhood base]] of compact neighbourhoods, and since $X$ is assumed to be Hausdorff, this implies all the other variants of definition of local compactness, by \href{locally+compact+topological+space#InHausdorffSpacesDefinitionsOfLocalCompactnessAgree}{this prop.}.

\textbf{1) $\Rightarrow$ 2)}

Let $X$ be sigma-compact. We show that then $X$ is [[second-countable topological space|second-countable]]:

By sigma-compactness there exists a [[countable set]] $\{K_i \subset X\}_{i \in I}$ of compact subspaces. By $X$ being locally Euclidean, each admits an [[open cover]] by restrictions of [[Euclidean spaces]]. By their compactness, each of these has a subcover $\{ \mathbb{R}^n \overset{\phi_{i,j}}{\to} X \}_{j \in J_i}$ with $J_i$ a finite set. Since [[countable unions of countable sets are countable]], we have obtained a countable cover by Euclidean spaces $\{ \mathbb{R}^n  \overset{\phi_{i,j}}{\to} X\}_{i \in I, j \in J_i}$. Now Euclidean space itself is second countable (by \href{second-countable+space#SecondCountableEuclideanSpace}{this example}), hence admits a countable set $\beta_{\mathbb{R}^n}$ of base open sets. As a result the union $\underset{{i \in I} \atop {j \in J_i}}{\cup} \phi_{i,j}(\beta_{\mathbb{R}^n})$ is a base of opens for $X$. But this is a countable union of countable sets, and since [[countable unions of countable sets are countable]] we have obtained a countable base for the topology of $X$. This means that $X$ is second-countable.

\textbf{1) $\Rightarrow$ 3)}

Let $X$ be sigma-compact. We show that then $X$ is paracompact with a countable set of connected components:

Since [[locally compact and sigma-compact spaces are paracompact]], it follows that $X$ is paracompact. By [[locally connected topological space|local connectivity]] (prop. \ref{LocalPropertiesOfLocallyEuclideanSpace}) $X$ is the [[disjoint union space]] of its [[connected components]] (\href{locally+connected+topological+space#AlternativeCharacterizationsOfLocalConnectivity}{this prop.}). Since, by the previous statement, $X$ is also second-countable it cannot have an uncountable set of connected components.

\textbf{2)$\Rightarrow$ 1)} Let $X$ be second-countable, we need to show that it is sigma-compact.

This follows since [[locally compact and second-countable spaces are sigma-compact]].

\textbf{3) $\Rightarrow$ 1)}

Now let $X$ be paracompact with countably many connected components. We show that $X$ is sigma-compact.

Since $X$ is locally compact, there exists a cover $\{K_i = Cl(U_i) \subset X\}_{i \in I}$ by [[compact topological space|compact]] [[subspaces]]. By paracompactness there is a locally finite refinement of this cover. Since [[paracompact Hausdorff spaces are normal]], the [[shrinking lemma]] applies to this refinement and yields a locally finite open cover

\begin{displaymath}
\mathcal{V} \coloneqq \{V_j \subset X \}_{j \in J}
\end{displaymath}
as well as a locally finite cover $\{Cl(V_j) \subset X\}_{j \in J}$ by closed subsets. Since this is a refinement of the orignal cover, all the $Cl(V_j)$ are contained in one of the compact subspaces $K_i$. Since [[subsets are closed in a closed subspace precisely if they are closed in the ambient space]], the $Cl(V_j)$ are also closed as subsets of the $K_i$. Since [[closed subsets of compact spaces are compact]] it follows that the $Cl(V_j)$ are themselves compact and hence form a locally finite cover by compact subspaces.

Now fix any $j_0 \in J$.

We claim that for every $j \in J$ there is a finite sequence of indices $(j_0, j_1, \cdots, j_n = j)$ with the property that $V_{j_k} \cap V_{j_{k+1}} \neq \emptyset$.

To see this, first observe that it is sufficient to show sigma-compactness for the case that $X$ is [[connected topological space|connected]]. From this the general statement follows since [[countable unions of countable sets are countable]]. Hence assume that $X$ is connected. It follows from lemma \ref{PathConnectedFromConnectedLocallyEuclideanSpace} that $X$ is [[path-connected topological space|path-connected]].

Hence for any $x \in V_{j_0}$ and $y \in V_{j}$ there is a path $\gamma \colon [0,1] \to X$ connecting $x$ with $y$. Since the [[closed interval]] is compact and since [[continuous images of compact spaces are compact]], it follows that there is a finite subset of the $V_i$ that covers the image of this path. This proves the claim.

It follows that there is a function

\begin{displaymath}
f \;\colon\; \mathcal{V} \longrightarrow \mathbb{N}
\end{displaymath}
which sends each $V_j$ to the [[minimum]] natural number as above.

We claim now that for all $n \in \mathbb{N}$ the [[preimage]] of $\{0,1, \cdots, n\}$ under this function is a [[finite set]]. Since [[countable unions of countable sets are countable]] this implies that $\{ Cl(V_j) \subset X\}_{j \in J}$ is a countable cover of $X$ by compact subspaces, hence that $X$ is sigma-compact.

We prove this last claim by [[induction]]. It is true for $n = 0$ by construction. Assume it is true for some $n \in \mathbb{N}$, hence that $f^{-1}(\{0,1, \cdots, n\})$ is a finite set. Since finite unions of compact subspaces are again compact (\href{compact+space#UnionsAndIntersectionOfCompactSubspaces}{this prop.}) it follows that

\begin{displaymath}
K_n
    \coloneqq
  \underset{V \in f^{-1}(\{0,\cdots, n\})}{\cup} V
\end{displaymath}
is compact. By local finiteness of the $\{V_j\}_{j \in J}$, every point $x \in K_n$ has an open neighbourhood $W_x$ that intersects only a finite set of the $V_j$. By compactness of $K_n$, the cover $\{W_x \subset X\}_{x \in K_n}$ has a finite subcover. In conclusion this implies that only a finite number of the $V_j$ intersect $K_n$.

Now by definition $f^{-1}(\{0,1,\cdots, n+1\})$ is a subset of those $V_j$ which intersect $K_n$, and hence itself finite.

\end{proof}
\hypertarget{topological_manifold}{}\subsubsection*{{Topological manifold}}\label{topological_manifold}

\begin{defn}
\label{TopologicalManifold}\hypertarget{TopologicalManifold}{}
\textbf{([[topological manifold]])}

A \emph{topological manifold is a [[topological space]] which is}

\begin{enumerate}%
\item [[locally Euclidean topological space|locally Euclidean]] (def. \ref{LocallyEuclideanSpace}),


\item [[paracompact Hausdorff topological space|paracompact Hausdorff]].



\end{enumerate}
If the local [[Euclidean spaces|Euclidean]] neighbourhoods $\mathbb{R}^n \overset{\simeq}{\to} U \subset X$ are all of [[dimension]] $n$ for a fixed $n \in \mathbb{N}$, then the topological manifold is said to be a \emph{$n$-dimensional manifold} or \emph{$n$-fold}. This is usually assumed to be the case.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
\textbf{(varying terminology)}

Often a topological manifold (def. \ref{TopologicalManifold}) is required to be [[sigma-compact]]. But by prop. \ref{RegularityConditionsForTopologicalManifoldsComparison} this is not an extra condition as long as there is a [[countable set]] of [[connected components]].

\end{remark}
\hypertarget{differentiable_manifolds}{}\subsubsection*{{Differentiable manifolds}}\label{differentiable_manifolds}

\begin{defn}
\label{Charts}\hypertarget{Charts}{}
\textbf{([[local chart]] and [[atlas]] and [[gluing function]])}

Given an $n$-dimensional topological manifold $X$ (def. \ref{TopologicalManifold}), then

\begin{enumerate}%
\item an [[open subset]] $U \subset X$ and a [[homeomorphism]] $\phi \colon \mathbb{R}^n \overset{\phantom{A}\simeq\phantom{A}}{\to} U$ is also called a \emph{[[local coordinate chart]]} of $X$.


\item an [[open cover]] of $X$ by local charts $\left\{ \mathbb{R}^n \overset{\phi_i}{\to} U \subset X \right\}_{i \in I}$ is called an \emph{[[atlas]]} of the topological manifold.


\item denoting for each $i,j \in I$ the [[intersection]] of the $i$th chart with the $j$th chart in such an atlas by

\begin{displaymath}
U_{i j} \coloneqq U_i \cap U_j
\end{displaymath}
then the induced homeomorphism

\begin{displaymath}
\mathbb{R}^n
    \supset
     \phantom{AA}
  \phi_i^{-1}(U_{i j})
    \overset{\phantom{A}\phi_i\phantom{A}}{\longrightarrow}
  U_{i j}
    \overset{\phantom{A}\phi_j^{-1}\phantom{A}}{\longrightarrow}
  \phi_j^{-1}(U_{i j})
    \phantom{AA}
    \subset
  \mathbb{R}^n
\end{displaymath}
is called the \emph{[[gluing function]]} from chart $i$ to chart $j$.



\end{enumerate}
\begin{quote}%
graphics grabbed from [[The Geometry of Physics - An Introduction|Frankel]]


\end{quote}
\end{defn}
\begin{defn}
\label{Differentiable}\hypertarget{Differentiable}{}
\textbf{([[differentiable manifold|differentiable]] and [[smooth manifolds]])}

For $p \in \mathbb{N} \cup \{\infty\}$ then a $p$-fold \emph{[[differentiable manifold]]} is

\begin{enumerate}%
\item a [[topological manifold]] $X$ (def. \ref{TopologicalManifold});


\item an [[atlas]] $\{\mathbb{R}^n \overset{\phi_i}{\to} X\}$ (def. \ref{Charts}) all whose [[gluing functions]] are $p$ times continuously [[differentiable function|differentiable]].



\end{enumerate}
A $p$-fold [[differentiable function]] between $p$-fold differentiable manifolds

\begin{displaymath}
(X, \{\mathbb{R}^{n} \overset{\phi_i}{\to} U_i \subset X\}_{i \in I})
   \overset{\phantom{AA}f\phantom{AA}}{\longrightarrow}
  (Y, \{\mathbb{R}^{n'} \overset{\psi_j}{\to} V_j \subset Y\}_{j \in J})
\end{displaymath}
is

\begin{itemize}%
\item a [[continuous function]] $f \colon X \to Y$

\end{itemize}
such that

\begin{itemize}%
\item for all $i \in I$ and $j \in J$ then

\begin{displaymath}
\mathbb{R}^n
   \supset
    \phantom{AA}
   (f\circ \phi_i)^{-1}(V_j)
     \overset{\phi_i}{\longrightarrow}
   f^{-1}(V_j)
    \overset{f}{\longrightarrow}
   V_j
     \overset{\psi_j^{-1}}{\longrightarrow}
   \mathbb{R}^{n'}
\end{displaymath}
is a $p$-fold [[differentiable function]] between open subsets of [[Euclidean space]].



\end{itemize}
\end{defn}
Notice that this in in general a non-trivial condition even if $X = Y$ and $f$ is the identity function. In this case the above exhibits a passage to a different, but equivalent, differentiable atlas.

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

\begin{prop}
\label{OpenSubsetsOfDifferentiableManifoldsAreDifferentiableManifolds}\hypertarget{OpenSubsetsOfDifferentiableManifoldsAreDifferentiableManifolds}{}
Let $X$ be a $k$-fold differentiable manifold and let $S \subset X$ be an [[open subset]] of the underlying [[topological space]] $(X,\tau)$.

Then $S$ carries the structure of a $k$-fold differentiable manifold such that the inclusion map $S \hookrightarrow X$ is an [[open embedding|open]] [[embedding of differentiable manifolds]].

\end{prop}
\begin{proof}
Since the underlying [[topological space]] of $X$ is [[locally connected topological space|locally connected]] (\href{topological+manifold#LocalPropertiesOfLocallyEuclideanSpace}{this prop.}) it is the [[disjoint union space]] of its [[connected components]] (\href{locally+connected+topological+space#AlternativeCharacterizationsOfLocalConnectivity}{this prop.}).

Therefore we are reduced to showing the statement for the case that $X$ has a single [[connected component]]. By \href{topological+manifold#RegularityConditionsForTopologicalManifoldsComparison}{this prop} this implies that $X$ is [[second-countable topological space]].

Now a [[subspace]] of a second-countable Hausdorff space is clearly itself second countable and Hausdorff.

Similarly it is immediate that $S$ is still [[locally Euclidean space|locally Euclidean]]: since $X$ is locally Euclidean every point $x \in S \subset X$ has a Euclidean neighbourhood in $X$ and since $S$ is open there exists an open ball in that (itself [[homeomorphism|homeomorphic]] to Euclidean space) which is a Euclidean neighbourhood of $x$ contained in $S$.

For the differentiable structure we pick these Euclidean neighbourhoods from the given atlas. Then the [[gluing functions]] for the Euclidean charts on $S$ are $k$-fold differentiable follows since these are restrictions of the gluing functions for the atlas of $X$.

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

See the examples at \emph{[[differentiable manifold]]}.

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

Textbook accounts include

\begin{itemize}%
\item John Lee, \emph{Introduction to topological manifolds}, Graduate Texts in Mathematics, Springer (2000) (errata \href{https://sites.math.washington.edu/~lee/Books/ITM/errata.pdf}{pdf})

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Topological_manifold}{Topological manifold}}

\end{itemize}
[[!redirects topological manifolds]]

[[!redirects locally Euclidean topological space]] [[!redirects locally Euclidean topological spaces]]

[[!redirects locally Euclidean space]] [[!redirects locally Euclidean spaces]]



\end{document}