Replaced references to upward-directed families with ideals.
This commit is contained in:
Bokuan Li
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src/topology/functions/ideal.tex
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52
src/topology/functions/ideal.tex
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\section{Ideals}
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\label{section:set-ideal}
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\begin{definition}[Ideal]
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\label{definition:set-ideal}
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Let $X$ be a set and $\sigma \subset 2^X$, then $\sigma$ is an \textbf{ideal} over $X$ if:
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\begin{enumerate}[label=(I\arabic*)]
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\item For any $E \in \sigma$ and $F \subset E$, $F \in \sigma$.
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\item For any $E, F \in \sigma$, $E \cup F \in \sigma$.
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\end{enumerate}
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\end{definition}
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\begin{definition}[Covering]
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\label{definition:set-ideal-covering}
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Let $X$ be a set and $\sigma \subset 2^X$, then $\sigma$ \textbf{covers} $X$/is \textbf{covering} if $\bigcup_{E \in \sigma}E = X$.
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\end{definition}
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\begin{definition}[Fundamental]
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\label{definition:set-ideal-fundamental}
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Let $X$ be a set, $\sigma \subset 2^X$ be an ideal, and $\tau \subset \sigma$, then $\tau$ is \textbf{fundamental} with respect to $\sigma$ if for any $E \in \sigma$, there exists $F \in \tau$ such that $E \subset F$.
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\end{definition}
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\begin{proposition}
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\label{proposition:set-ideal-fundamental-criterion}
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Let $X$ be a set and $\tau \subset 2^X$, then the following are equivalent:
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\begin{enumerate}
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\item For any $E, F \in \tau$, there exists $G \in \tau$ such that $E \cup F \subset G$.
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\item There exists an ideal $\sigma \subset 2^X$ such that $\tau$ is fundamental with respect to $\sigma$.
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\end{enumerate}
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If the above holds, then the ideal $\sigma$ in (2) is the \textbf{ideal generated by} $\tau$.
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\end{proposition}
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\begin{proof}
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(1) $\Rightarrow$ (2): Let
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\[
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\sigma = \bracs{F \subset X| \exists E \in \tau: F \subset E}
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\]
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then $\sigma$ satisfies (I1) by definition. For any $E, F \in \sigma$, there exists $E_0, F_0 \in \tau$ such that $E \subset E_0$ and $F \subset F_0$. By assumption, there exists $G \in \tau$ such that
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\[
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E \cup F \subset E_0 \cup F_0 \subset G
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\]
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so $\sigma$ satisfies (I2), and is an ideal.
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(2) $\Rightarrow$ (1): Let $E, F \in \tau$, then $E \cup F \in \sigma$. Since $\tau$ is fundamental, there exists $G \in \tau$ such that $E \cup F \subset G$.
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\end{proof}
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@@ -1,5 +1,6 @@
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\chapter{Function Spaces}
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\label{chap:function-spaces}
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\input{./ideal.tex}
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\input{./set-systems.tex}
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\input{./uniform.tex}
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@@ -1,58 +1,58 @@
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\section{Topology With Respect to Families of Sets}
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\section{Topology With Respect to Ideals}
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\label{section:pointwise}
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\begin{definition}[Set-Open Topology]
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\label{definition:set-open}
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Let $T$ be a set, $\mathfrak{S} \subset 2^T$ be a non-empty family of sets, and $(X, \topo)$ be a topological space. For each $S \in \mathfrak{S}$ and $U \subset X$ open, let
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Let $T$ be a set, $\sigma \subset 2^T$ be an ideal, and $(X, \topo)$ be a topological space. For each $S \in \sigma$ and $U \subset X$ open, let
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\[
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M(S, U) = \bracs{f \in X^T| f(S) \subset U}
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\]
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and
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\[
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\ce(\mathfrak{S}, \topo) = \bracs{M(S, U)| S \in \mathfrak{S}, U \in \topo}
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\ce(\sigma, \topo) = \bracs{M(S, U)| S \in \sigma, U \in \topo}
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\]
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then the topology generated by $\ce$ is the \textbf{$\mathfrak{S}$-open topology} on $T^X$.
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then the topology generated by $\ce$ is the \textbf{$\sigma$-open topology} on $T^X$.
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If $\cb \subset \topo$ generates $\topo$, then $\ce(\mathfrak{S}, \cb)$ generates the $\mathfrak{S}$-open topology.
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If $\cb \subset \topo$ generates $\topo$, then $\ce(\sigma, \cb)$ generates the $\sigma$-open topology.
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\end{definition}
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\begin{definition}[Set Uniformity]
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\label{definition:set-uniform}
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Let $T$ be a set, $\mathfrak{S} \subset 2^T$ be a non-empty family of sets, and $(X, \fU)$ be a uniform space. For each $S \in \mathfrak{S}$ and $U \in \fU$, let
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Let $T$ be a set, $\sigma \subset 2^T$ be an ideal, and $(X, \fU)$ be a uniform space. For each $S \in \sigma$ and $U \in \fU$, let
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\[
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E(S, U) = \bracs{(f, g) \in X^T|(f(x), g(x)) \in U \forall x \in S}
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\]
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and
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\[
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\mathfrak{E}(\mathfrak{S}, \fU) = \bracs{E(S, U)| S \in \mathfrak{S}, U \in \fU}
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\mathfrak{E}(\sigma, \fU) = \bracs{E(S, U)| S \in \sigma, U \in \fU}
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\]
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then
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\begin{enumerate}
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\item $\mathfrak{E}(\mathfrak{S}, \fU)$ generates a uniformity $\fV$ on $X^T$.
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\item The topology induced by $\fV$ is finer than the $\mathfrak{S}$-topology on $T^X$.
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\item If $\mathfrak{S}$ is upward-directed with respect to inclusion, then $\mathfrak{E}(\mathfrak{S}, \fU)$ is forms a fundamental system of entourages for $\fV$.
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\item $\mathfrak{E}(\sigma, \fU)$ generates a uniformity $\fV$ on $X^T$.
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\item The topology induced by $\fV$ is finer than the $\sigma$-open topology on $T^X$.
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\item If $\mathfrak{E}(\sigma, \fU)$ forms a fundamental system of entourages for $\fV$.
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\end{enumerate}
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The uniformity $\fV$ is the \textbf{$\mathfrak{S}$-uniformity}, and the topology induced by $\fV$ is the \textbf{topology of uniform convergence on the sets $\mathfrak{S}$}/\textbf{$\mathfrak{S}$-uniform topology} on $X^T$.
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The uniformity $\fV$ is the \textbf{$\sigma$-uniformity}, and the topology induced by $\fV$ is the \textbf{topology of uniform convergence on the sets $\sigma$}/\textbf{$\sigma$-uniform topology} on $X^T$.
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\end{definition}
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\begin{proof}
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(1): Since $\Delta \subset E(S, U)$ for all $S \in \mathfrak{S}$ and $U \in \fU$, $\mathfrak{E}(\mathfrak{S}, \fU)$ generates a uniformity on $X^T$.
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(1): Since $\Delta \subset E(S, U)$ for all $S \in \sigma$ and $U \in \fU$, $\mathfrak{E}(\sigma, \fU)$ generates a uniformity on $X^T$.
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(2): Let $U \subset X$ be open, then for each $x \in U$, there exists $V_x \in \fU$ such that $x \in V_x(x) \subset U$. In which case, $U = \bigcup_{x \in U}V_x(x)$ and $M(S, U) = \bigcup_{x \in U}M(S, V_x)(x)$.
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(3): It is sufficient to verify
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\begin{enumerate}
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\item[(FB1)] For any $S, S' \in \mathfrak{S}$, there exists $T \in \mathfrak{S}$ with $T \supset S, S'$. In which case, for any $U, U' \in \fU$,
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\item[(FB1)] For any $S, S' \in \sigma$, there exists $T \in \sigma$ with $T \supset S, S'$. In which case, for any $U, U' \in \fU$,
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\[
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E(T, U \cap U') \subset E(S \cup S', U \cap U') \subset E(S, U) \cap E(S', U')
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\]
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\item[(UB1)] For any $U \in \fU$, $\Delta \subset U$. Thus the diagonal in $X^T$ is in $E(S, U)$ for any $S \in \mathfrak{S}$.
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\item[(UB2)] For any $U \in \fU$, there exists $V \in \fV$ with $V \circ V \subset U$. Thus for any $S \in \mathfrak{S}$,
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\item[(UB1)] For any $U \in \fU$, $\Delta \subset U$. Thus the diagonal in $X^T$ is in $E(S, U)$ for any $S \in \sigma$.
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\item[(UB2)] For any $U \in \fU$, there exists $V \in \fV$ with $V \circ V \subset U$. Thus for any $S \in \sigma$,
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\[
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E(S, V) \circ E(S, V) \subset E(S, V \circ V) \subset E(S, U)
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\]
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@@ -63,43 +63,43 @@
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\begin{proposition}
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\label{proposition:set-uniform-pseudometric}
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Let $T$ be a set, $\mathfrak{S} \subset 2^T$ be a non-empty family of sets, and $(X, \fU)$ be a uniform space whose uniformity is induced by the pseudometrics $\seqi{d}$. For each $i \in I$ and $S \in \mathfrak{S}$, let
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Let $T$ be a set, $\sigma \subset 2^T$ be an ideal, and $(X, \fU)$ be a uniform space whose uniformity is induced by the pseudometrics $\seqi{d}$. For each $i \in I$ and $S \in \sigma$, let
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\[
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d_{i, S}: X^T \times X^T \quad (f, g) \mapsto \sup_{x \in S}d_i(f(x), g(x))
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\]
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then
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\begin{enumerate}
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\item $\bracs{d_{i, S}| i \in I, S \in \mathfrak{S}}$ is a family of pseudometrics induces the $\mathfrak{S}$-uniformity on $X^T$.
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\item If $\mathfrak{S}$ is upward-directed with respect to inclusion, then
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\item $\bracs{d_{i, S}| i \in I, S \in \sigma}$ is a family of pseudometrics induces the $\sigma$-uniformity on $X^T$.
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\item The family
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\[
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\bracs{\bigcap_{j \in J}E(d_{j, S}, r)|J \subset I \text{ finite}, r > 0, S \in \mathfrak{S}}
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\bracs{\bigcap_{j \in J}E(d_{j, S}, r)|J \subset I \text{ finite}, r > 0, S \in \sigma}
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\]
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is a fundamental system of entourages for the $\mathfrak{S}$-uniformity on $X^T$.
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is a fundamental system of entourages for the $\sigma$-uniformity on $X^T$.
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\end{enumerate}
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\end{proposition}
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\begin{proof}
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(1): Let $S \in \mathfrak{S}$ and $U \in \fU$, then there exists $r > 0$ and $J \subset I$ finite such that $\bigcap_{j \in J}E(d_j, r) \subset U$, so
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(1): Let $S \in \sigma$ and $U \in \fU$, then there exists $r > 0$ and $J \subset I$ finite such that $\bigcap_{j \in J}E(d_j, r) \subset U$, so
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\[
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\bigcap_{j \in J}E(d_{j, S}, r) \subset E\paren{S, \bigcap_{j \in J}E(d_j, r)} \subset E(S, U)
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\]
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and the uniformity induced by $\bracs{d_{i, S}| i \in I, S \in \mathfrak{S}}$ contains the $\mathfrak{S}$-uniformity.
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and the uniformity induced by $\bracs{d_{i, S}| i \in I, S \in \sigma}$ contains the $\sigma$-uniformity.
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On the other hand, for any $i \in I$ and $r > 0$, $E(d_j, r/2) \in \fU$ by \autoref{definition:pseudometric-uniformity}. Therefore $E(S, E(d_j, r/2)) \subset E(d_{j, S}, r)$, so the $\mathfrak{S}$-uniformity contains the induced uniformity.
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On the other hand, for any $i \in I$ and $r > 0$, $E(d_j, r/2) \in \fU$ by \autoref{definition:pseudometric-uniformity}. Therefore $E(S, E(d_j, r/2)) \subset E(d_{j, S}, r)$, so the $\sigma$-uniformity contains the induced uniformity.
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(2): If $\mathfrak{S}$ is upward-directed with respect to inclusion, then by \autoref{definition:set-uniform},
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(2): By \autoref{definition:set-uniform},
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\[
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\bracs{E(S, U)| U \in \fU, S \in \mathfrak{S}}
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\bracs{E(S, U)| U \in \fU, S \in \sigma}
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\]
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Following the same steps in (1),
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is a fundamental system of entourages for the $\sigma$-uniformity. Following the same steps in (1),
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\[
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\bracs{\bigcap_{j \in J}E(d_{j, S}, r)|J \subset I \text{ finite}, r > 0, S \in \mathfrak{S}}
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\bracs{\bigcap_{j \in J}E(d_{j, S}, r)|J \subset I \text{ finite}, r > 0, S \in \sigma}
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\]
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is a fundamental system of entourages for the $\mathfrak{S}$-uniformity.
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is a fundamental system of entourages for the $\sigma$-uniformity.
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\end{proof}
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@@ -109,7 +109,7 @@
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Let $T$ be a set and $X$ be a topological space, then the following topologies on $X^T$ coincide:
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\begin{enumerate}
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\item The product topology on $X^T$.
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\item The $\mathfrak{S}$-open topology, where $\mathfrak{S} = \bracs{\bracs{x}| x \in X}$.
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\item The $\sigma$-open topology, where $\sigma$ is the collection of all finite sets.
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\item (If $X$ is a uniform space) The $\mathfrak{F}$-uniform topology, where $\fF = \bracs{F| F \subset X \text{ finite}}$.
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\end{enumerate}
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This topology is the \textbf{topology of pointwise convergence} on $X^T$.
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@@ -125,10 +125,10 @@
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\begin{proposition}
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\label{proposition:set-uniform-complete}
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Let $T$ be a set, $\mathfrak{S} \subset 2^T$ such that $\bigcup_{S \in \mathfrak{S}}S = T$, and $(X, \fU)$ be a complete uniform space, then $X^T$ equipped with the $\mathfrak{S}$-uniformity is complete.
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Let $T$ be a set, $\sigma \subset 2^T$ be a covering ideal, and $(X, \fU)$ be a complete uniform space, then $X^T$ equipped with the $\sigma$-uniformity is complete.
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\end{proposition}
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\begin{proof}
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Let $\fF \subset 2^{X^T}$ be a Cauchy filter. Let $x \in T$. If there exists $S \in \mathfrak{S}$ with $x \in S$, then $\pi_x(\fF) \subset 2^X$ is a Cauchy filter. By completeness of $X$, there exists $f: T \to X$ such that $\pi_x(\fF) \to f(x)$ for all $x \in \bigcup_{S \in \mathfrak{S}}S$.
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Let $\fF \subset 2^{X^T}$ be a Cauchy filter. Let $x \in T$. If there exists $S \in \sigma$ with $x \in S$, then $\pi_x(\fF) \subset 2^X$ is a Cauchy filter. By completeness of $X$, there exists $f: T \to X$ such that $\pi_x(\fF) \to f(x)$ for all $x \in \bigcup_{S \in \sigma}S$.
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Let $S \in \mathfrak{S}$ be non-empty and $V_0, V$ be symmetric entourages of $X$ with $V_0 \circ V_0 \subset V$. Since $\fF$ is Cauchy, there exists an $E(S, V_0)$-small set $F \in \fF$. Let $x \in S$, then $\pi_x(\fF) \to f(x)$ implies that $F_x = \pi_x^{-1}(V_0(f(x))) \in \fF$. Since $F \cap F_x \ne \emptyset$, $(f(x), g(x)) \in V_0 \circ V_0 \subset V$ for all $g \in F$. As this holds for all $x \in S$, $E \subset E(S, V)(f) \in \fF$. Thus $\fF \to f$.
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Let $S \in \sigma$ be non-empty and $V_0, V$ be symmetric entourages of $X$ with $V_0 \circ V_0 \subset V$. Since $\fF$ is Cauchy, there exists an $E(S, V_0)$-small set $F \in \fF$. Let $x \in S$, then $\pi_x(\fF) \to f(x)$ implies that $F_x = \pi_x^{-1}(V_0(f(x))) \in \fF$. Since $F \cap F_x \ne \emptyset$, $(f(x), g(x)) \in V_0 \circ V_0 \subset V$ for all $g \in F$. As this holds for all $x \in S$, $E \subset E(S, V)(f) \in \fF$. Thus $\fF \to f$.
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\end{proof}
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@@ -17,7 +17,7 @@
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$U \circ V$ & Composition of $U, V \subset X \times X$. & \autoref{definition:composition} \\
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$U(A)$ & Slice of $U \subset X \times Y$ at $A \subset X$: $\{y \mid \exists x \in A,\, (x,y) \in U\}$. & \autoref{definition:slice} \\
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$E(S, U)$ & Entourage of the form $\{(f,g) \in X^T \mid (f(x),g(x)) \in U\ \forall x \in S\}$. & \autoref{definition:set-uniform} \\
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$\mathfrak{E}(\mathfrak{S}, \mathfrak{U})$ & $\mathfrak{S}$-uniformity, generated by $\{E(S,U) \mid S \in \mathfrak{S},\ U \in \mathfrak{U}\}$. & \autoref{definition:set-uniform} \\
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$\mathfrak{E}(\sigma, \mathfrak{U})$ & $\sigma$-uniformity, generated by $\{E(S,U) \mid S \in \sigma,\ U \in \mathfrak{U}\}$. & \autoref{definition:set-uniform} \\
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% Function Spaces
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$\mathrm{supp}(f)$ & Support of $f$. & \autoref{definition:support} \\
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