From 28407333ffceca9b99fae721c30e8ae146a863da Mon Sep 17 00:00:00 2001 From: Justin Gassner Date: Wed, 14 Feb 2024 07:24:38 +0100 Subject: Update --- .../general-topology/continuity-and-convergence.md | 24 ++++++++++++++++++++++ 1 file changed, 24 insertions(+) create mode 100644 pages/general-topology/continuity-and-convergence.md (limited to 'pages/general-topology/continuity-and-convergence.md') diff --git a/pages/general-topology/continuity-and-convergence.md b/pages/general-topology/continuity-and-convergence.md new file mode 100644 index 0000000..7ae4534 --- /dev/null +++ b/pages/general-topology/continuity-and-convergence.md @@ -0,0 +1,24 @@ +--- +title: Continuity & Convergence +parent: General Topology +nav_order: 2 +--- + +# {{ page.title }} + +{% definition Continuity %} +A mapping $f: X \to Y$ between topological spaces $X$ and $Y$ is called *continuous*, +if for each open subset $V$ of $Y$ the inverse image $f^{-1}(V)$ is an open subset of $X$. +{% enddefinition %} + +Slogan: continuous $=$ The inverse image of every open subset is open. + +{% definition Homeomorphism %} +Suppose $X$ and $Y$ are topological spaces. +A mapping $f: X \to Y$ is said to be a *homeomorphism*, +if $f$ is bijective and both $f$ and the inverse mapping $f^{-1} : Y \to X$ are continuous. +{% enddefinition %} + +## Nets + +## Filters -- cgit v1.2.3-70-g09d2