From 7c66b227a494748e2a546fb85317accd00aebe53 Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Thu, 15 Feb 2024 05:11:07 +0100
Subject: Update

---
 pages/general-topology/metric-spaces/index.md | 24 ++++++++++++++----------
 1 file changed, 14 insertions(+), 10 deletions(-)

(limited to 'pages/general-topology/metric-spaces/index.md')

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 %}
 
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