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Book keeping.

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Bokuan Li
2026-06-29 16:34:06 -04:00
parent 65a2b4cef4
commit a11cfe4e04
4 changed files with 13 additions and 11 deletions
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src/topology/main/preimage.tex
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@@ -1,7 +1,7 @@
\section{Open Preimage Functions}
\label{section:preimage-function-topology}
\begin{definition}[Open Preimage Function]
\begin{definition}[Open Preimage Function*]
\label{definition:open-preimage-function}
Let $X$ be a set, $(Y, \topo)$ be a topological space, and $P: \topo \to 2^X$, then $P$ is an \textbf{open preimage function} if
\begin{enumerate}[label=(PF\arabic*)]
@@ -15,7 +15,7 @@
\label{proposition:open-preimage-function-gymnastics}
Let $X$ be a set, $(Y, \topo)$ be a topological space, and $f: X \to Y$, then:
\begin{enumerate}
\item The mapping $U \mapsto f^{-1}(U)$ is an open preimage function.
\item The mapping $U \mapsto f^{-1}(U)$ is an \hyperref[open preimage function]{definition:open-preimage-function}.
\item If $Y$ is T1, then for any $g: X \to Y$ with $g^{-1}(U) = f^{-1}(U)$ for all $U \in \topo$, $f = g$.
\end{enumerate}
\end{proposition}
@@ -29,7 +29,7 @@
\end{proof}
\begin{definition}[Basic Preimage Function]
\begin{definition}[Basic Preimage Function*]
\label{definition:basic-preimage-function}
Let $X$ be a set, $Y$ be a topological space, $\mathcal{B}$ be a base for $Y$ with $\emptyset \in \mathcal{B}$, and $p: \mathcal{B} \to 2^X$, then $p$ is a \textbf{basic preimage function} if:
\begin{enumerate}[label=(PF\arabic*)]
@@ -43,8 +43,8 @@
\label{proposition:basic-preimage-function}
Let $X$ be a set, $(Y, \topo)$ be a topological space, and $\mathcal{B}$ be a base for $Y$ with $\emptyset \in \mathcal{B}$, then:
\begin{enumerate}
\item For any open preimage function $P: \topo \to 2^X$, $P|_{\mathcal{B}}$ is a basic preimage function.
\item For any basic preimage function $p: \mathcal{B} \to 2^X$, there exists a unique open preimage function $P: \topo \to 2^X$ such that $p = P|_{\mathcal{B}}$.
\item For any \hyperref[open preimage function]{definition:open-preimage-function} $P: \topo \to 2^X$, $P|_{\mathcal{B}}$ is a basic preimage function.
\item For any \hyperref[basic preimage function]{definition:basic-preimage-function} $p: \mathcal{B} \to 2^X$, there exists a unique open preimage function $P: \topo \to 2^X$ such that $p = P|_{\mathcal{B}}$.
\end{enumerate}
\end{proposition}
\begin{proof}
@@ -84,7 +84,7 @@
\begin{theorem}
\label{theorem:open-preimage-function-existence}
Let $X$ be a set, $(Y, \fU)$ be a complete Hausdorff uniform space with topology $\topo$, and $P: \topo \to 2^X$ be an open preimage function such that:
Let $X$ be a set, $(Y, \fU)$ be a complete Hausdorff uniform space with topology $\topo$, and $P: \topo \to 2^X$ be an \hyperref[open preimage function]{definition:open-preimage-function} such that:
\begin{enumerate}
\item[(S)] For each $x \in X$ and $U \in \fU$, there exists a $U$-small set $V \in \topo$ such that $x \in P(V)$.
\end{enumerate}
@@ -106,7 +106,7 @@
\begin{corollary}
\label{corollary:basic-preimage-function-existence}
Let $X$ be a set, $(Y, \fU)$ be a complete Hausdorff uniform space, $\mathcal{B}$ be a base for $Y$ with $\emptyset \in \mathcal{B}$, and $p: \mathcal{B} \to 2^X$ be a basic preimage function such that:
Let $X$ be a set, $(Y, \fU)$ be a complete Hausdorff uniform space, $\mathcal{B}$ be a base for $Y$ with $\emptyset \in \mathcal{B}$, and $p: \mathcal{B} \to 2^X$ be a \hyperref[basic preimage function]{definition:basic-preimage-function} such that:
\begin{enumerate}
\item[(S')] For each $x \in X$ and $U \in \fU$, there exists a $U$-small set $V \in \mathcal{B}$ such that $x \in P(V)$.
\end{enumerate}