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authorJustin Gassner <justin.gassner@mailbox.org>2024-02-15 05:11:07 +0100
committerJustin Gassner <justin.gassner@mailbox.org>2024-02-15 05:11:07 +0100
commit7c66b227a494748e2a546fb85317accd00aebe53 (patch)
tree9c649667d2d024b90b32d36ca327ac4b2e7caeb2 /pages/general-topology/metric-spaces/index.md
parent28407333ffceca9b99fae721c30e8ae146a863da (diff)
downloadsite-7c66b227a494748e2a546fb85317accd00aebe53.tar.zst
Update
Diffstat (limited to 'pages/general-topology/metric-spaces/index.md')
-rw-r--r--pages/general-topology/metric-spaces/index.md24
1 files changed, 14 insertions, 10 deletions
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 %}