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, $(X,\mathcal{ J })$, 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 $X$.
  • If $X$ is not disconnected it is connected.

Then "we can characterise connectedness" by the familiar:

• $X$ is connected if and only if the only subsets of $X$ that are both open and closed are the empty set, $\emptyset$, and $X$ itself.

Leads to "main theorem on connectedness":

• $f:X\rightarrow Y$ cont., if $X$ connected then $f(X)$ 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...")