pages/general-topology/continuity-and-convergence.md


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---
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 %}

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