Notes:Connected space
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Contents
Overview
There are many equivalent definitions for connected. Here I attempt to document them, as research for the connectedness page.
Definitions
Introduction to Topological Manifolds
First:
- A topological space, [ilmath](X,\mathcal{ J })[/ilmath], is said to be disconnected if it can be expressed as the union of two disjoint and non-empty open sets[1].
Then "we can characterise connectedness" by the familiar:
- [ilmath]X[/ilmath] is connected if and only if the only subsets of [ilmath]X[/ilmath] that are both open and closed are the empty set, [ilmath]\emptyset[/ilmath], and [ilmath]X[/ilmath] itself[1].
Leads to "main theorem on connectedness":
- [ilmath]f:X\rightarrow Y[/ilmath] cont., if [ilmath]X[/ilmath] connected then [ilmath]f(X)[/ilmath] connected[1].
- Corollary: Every space homeomorphic to a connected space is connected[1].
Notes
I like this because it combines an intuitive definition with one involving open sets (rather than just "if it can be expressed...")
