Notes:Connected space

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.
  • Any such subsets are said to disconnect [ilmath]X[/ilmath].
  • If [ilmath]X[/ilmath] is not disconnected it is connected.

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.

Leads to "main theorem on connectedness":

• [ilmath]f:X\rightarrow Y[/ilmath] cont., if [ilmath]X[/ilmath] connected then [ilmath]f(X)[/ilmath] connected.
  • Corollary: Every space homeomorphic to a connected space is connected.

Notes

I like this because it combines an intuitive definition with one involving open sets (rather than just "if it can be expressed...")