From 7c66b227a494748e2a546fb85317accd00aebe53 Mon Sep 17 00:00:00 2001 From: Justin Gassner Date: Thu, 15 Feb 2024 05:11:07 +0100 Subject: Update --- pages/general-topology/compactness/index.md | 4 ++-- .../general-topology/continuity-and-convergence.md | 2 +- pages/general-topology/metric-spaces/index.md | 24 +++++++++++++--------- pages/general-topology/topological-spaces.md | 12 +++++------ 4 files changed, 23 insertions(+), 19 deletions(-) (limited to 'pages/general-topology') diff --git a/pages/general-topology/compactness/index.md b/pages/general-topology/compactness/index.md index 37e9b4d..6c2e274 100644 --- a/pages/general-topology/compactness/index.md +++ b/pages/general-topology/compactness/index.md @@ -26,7 +26,8 @@ if and only if it has the following property: then there exists a finite subcollection of $\mathcal{O}$ that covers $X$. If $\mathcal{A}$ is a collection of subsets of $X$, -let $\mathcal{A}^c = \braces{ X \setminus A : A \in \mathcal{A}}$ denote the collection of the complements of its members. +let $\mathcal{A}^c = \braces{ X \setminus A : A \in \mathcal{A}}$ denote +the collection of the complements of its members. Clearly, $\mathcal{B}$ is a subcollection of $\mathcal{A}$ if and only if $\mathcal{B}^c$ is a subcollection of $\mathcal{A}^c$. Moreover, note that $\mathcal{B}$ covers $X$ if and only if @@ -43,4 +44,3 @@ if and only if $\mathcal{A}^c$ consists of closed subsets of $X$. {% definition Finite Intersection Property%} TODO {% enddefinition %} - diff --git a/pages/general-topology/continuity-and-convergence.md b/pages/general-topology/continuity-and-convergence.md index 7ae4534..57e5ca9 100644 --- a/pages/general-topology/continuity-and-convergence.md +++ b/pages/general-topology/continuity-and-convergence.md @@ -1,5 +1,5 @@ --- -title: Continuity & Convergence +title: Continuity & Convergence parent: General Topology nav_order: 2 --- diff --git a/pages/general-topology/metric-spaces/index.md b/pages/general-topology/metric-spaces/index.md index c0dc45a..52b2b4c 100644 --- a/pages/general-topology/metric-spaces/index.md +++ b/pages/general-topology/metric-spaces/index.md @@ -46,13 +46,13 @@ Clearly, a metric subspace of a metric space is itself a metric space. {% proposition %} Let $(X,d)$ be a (semi-)metric space. - For all $x,y,z \in X$ we have the *inverse triangle inequality* - + $$ \abs{d(x,y) - d(y,z)} \le d(x,z). $$ - For all $v,w,x,y \in X$ we have the *quadrilateral inequality* - + $$ \abs{d(v,w) - d(x,y)} \le d(v,x) + d(w,y) $$ @@ -141,27 +141,31 @@ every sequence in $X$ has at most one limit. Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. A mapping $f: X \to Y$ is called - *continuous at a point $x \in X$* if - + $$ - \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x' \in X : \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen + \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x' \in X : + \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen $$ - *continuous* if it is continuous at every point of $X$, that is - + $$ - \forall x \in X \ \ \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x' \in X : \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen + \forall x \in X \ \ \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x' \in X : + \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen $$ - *uniformly continuous* if - + $$ - \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x,x' \in X : \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen + \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x,x' \in X : + \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen $$ - *Lipschitz continuous* if - + $$ - \exists L \ge 0 \ \ \forall x,x' \in X : d_Y(f(x),f(x')) \le L \, d_X(x,x') + \exists L \ge 0 \ \ \forall x,x' \in X : + d_Y(f(x),f(x')) \le L \, d_X(x,x') $$ {% enddefinition %} diff --git a/pages/general-topology/topological-spaces.md b/pages/general-topology/topological-spaces.md index b0b1834..cb0c30b 100644 --- a/pages/general-topology/topological-spaces.md +++ b/pages/general-topology/topological-spaces.md @@ -75,7 +75,8 @@ is the smallest topology on $X$ containing $\mathcal{A}$. {% definition Basis for a Topology %} A *basis for a topology* on a set $X$ is a collection $\mathcal{B}$ of subsets of $X$ -such that for every point $x \in X$ +such that for every point $x \in X$ + - there exists $B \in \mathcal{B}$ such that $x \in B$, - if $x \in B_1 \cap B_2$ for $B_1, B_2 \in \mathcal{B}$, then there exists a $B_3 \in \mathcal{B}$ @@ -85,6 +86,7 @@ such that for every point $x \in X$ {% theorem Topology Generated by a Basis %} If $X$ is set and $\mathcal{B}$ is a basis for a topology on $X$, then the topology generated by $\mathcal{B}$ equals + - the collection of all subsets $S \subset X$ with the property that for every $x \in S$ there exists a basis element $B \in \mathcal{B}$ such that $x \in B$ and $B \subset S$; @@ -125,7 +127,7 @@ then the topology generated by $\mathcal{S}$ equals Suppose $(X,\mathcal{T})$ is a topological space. A subset $S$ of $X$ is called *open* with respect to $\mathcal{T}$ -when it belongs to $\mathcal{T}$ +when it belongs to $\mathcal{T}$, and it is called *closed* with respect to $\mathcal{T}$ when its complement $X \setminus S$ belongs to $\mathcal{T}$. {% enddefinition %} @@ -137,10 +139,8 @@ if and only if its complement is closed. Let $\mathcal{C}$ be the collection of closed subsets of a topological space. Then {: .mb-0 } - $X$ and $\varnothing$ belong to $\mathcal{C}$, -- the intersection of any subcollection of $\mathcal{C}$ belongs to $\mathcal{C}$, -- the union of any finite subcollection $\mathcal{C}$ belongs to $\mathcal{C}$. +- the intersection of any subcollection of $\mathcal{C}$ belongs to $\mathcal{C}$, +- the union of any finite subcollection $\mathcal{C}$ belongs to $\mathcal{C}$. {% endproposition %} ## The Subspace Topology - - -- cgit v1.2.3-70-g09d2