Adjusted citation formats. Moved citation off of named theorems if possible.
All checks were successful
Compile Project / Compile (push) Successful in 22s
All checks were successful
Compile Project / Compile (push) Successful in 22s
This commit is contained in:
@@ -1,7 +1,7 @@
|
||||
\section{Regular Spaces}
|
||||
\label{section:regularspaces}
|
||||
|
||||
\begin{definition}[Regular Space, {{\cite[Proposition 1.4.11]{Bourbaki}}}]
|
||||
\begin{definition}[Regular Space]
|
||||
\label{definition:regular}
|
||||
Let $X$ be a topological space, then the following are equivalent:
|
||||
\begin{enumerate}
|
||||
@@ -10,7 +10,7 @@
|
||||
\end{enumerate}
|
||||
If $X$ is a T1 space such that the above holds, then $X$ is \textbf{regular}.
|
||||
\end{definition}
|
||||
\begin{proof}
|
||||
\begin{proof}[Proof {{\cite[Proposition 1.4.11]{Bourbaki}}}. ]
|
||||
$(1) \Rightarrow (2)$: Let $U \in \cn^o(x)$, then $U^c$ is closed with $x \not\in U^c$ by (V1). Thus there exists $V \in \cn^o(U^c)$ such that $x \not\in V$, so $V^c \in \cn(x)$ is closed with $V^c \subset U$.
|
||||
|
||||
$(2) \Rightarrow (1)$: Since $X$ is Hausdorff, $X$ is T1 as well. Let $x \in X$ and $A \subset X$ closed such that $x \not\in A$, then $A^c \in \cn(x)$. So there exists $K \in \cn(x)$ closed such that $x \in K \subset A^c$. Thus $K \in \cn(x)$ and $K^c \in \cn(A)$ are the desired neighbourhoods.
|
||||
|
||||
Reference in New Issue
Block a user