diff --git a/document.tex b/document.tex index 67b7cb2..7c02ece 100644 --- a/document.tex +++ b/document.tex @@ -6,6 +6,8 @@ Hi, welcome to my digital garden, where I collect math results that I learn. +Occasionally, I make up some definitions to play with. These definition blocks will always have a * at the end of its tital to indicate that it lives mostly in my head. These terms will always be referenced with a link to their definition block. + \input{./src/cat/index} \input{./src/topology/index} \input{./src/fa/index} diff --git a/src/cat/gluing/level.tex b/src/cat/gluing/level.tex index bfe8877..190bc37 100644 --- a/src/cat/gluing/level.tex +++ b/src/cat/gluing/level.tex @@ -1,7 +1,7 @@ \section{Preimages} \label{section:preimage} -\begin{definition}[Preimage Function] +\begin{definition}[Preimage Function*] \label{definition:preimage-function} Let $X, Y$ be sets and $P: 2^Y \to 2^X$, then $P$ is a \textbf{preimage function} if \begin{enumerate}[label=(PF\arabic*)] @@ -20,7 +20,7 @@ \label{proposition:preimage-gymnastics} Let $X$ and $Y$ be sets, then: \begin{enumerate} - \item For any $f: X \to Y$, the mapping $S \mapsto f^{-1}(S)$ is a total preimage function. + \item For any $f: X \to Y$, the mapping $S \mapsto f^{-1}(S)$ is a total \hyperref[preimage function]{definition:preimage-function}. \item For any total preimage function $P: 2^Y \to 2^X$, there exists a unique $f: X \to Y$ such that $P(S) = f^{-1}(S)$ for all $S \in 2^Y$. \end{enumerate} \end{proposition} diff --git a/src/measure/measurable-maps/approx.tex b/src/measure/measurable-maps/approx.tex index ed075f0..ee81e65 100644 --- a/src/measure/measurable-maps/approx.tex +++ b/src/measure/measurable-maps/approx.tex @@ -1,7 +1,7 @@ \section{Approximations with Simple Functions} \label{section:simple-approx} -\begin{definition}[Admissible Approximant Function] +\begin{definition}[Admissible Approximant Function*] \label{definition:admissible-approximant-function} Let $X$ be a topological space and $\mathcal{A}: X \to 2^X$, then $\mathcal{A}$ is an \textbf{admissible approximant function} on $X$ if: \begin{enumerate}[label=(AA\arabic*)] @@ -21,7 +21,7 @@ \end{lemma} -\begin{definition}[Approximation of the Identity] +\begin{definition}[Approximation of the Identity*] \label{definition:approximation-id-measure} Let $X$ be a topological space and $\net{I} \subset X^X$ be a net, then $\net{I}$ is an \textbf{approximation of the identity} if: \begin{enumerate}[label=(AI\arabic*)] diff --git a/src/topology/main/preimage.tex b/src/topology/main/preimage.tex index 35eb0a8..69209fb 100644 --- a/src/topology/main/preimage.tex +++ b/src/topology/main/preimage.tex @@ -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}