Notes:Connected space

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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].
    • Any such subsets are said to disconnect [ilmath]X[/ilmath][1].
    • If [ilmath]X[/ilmath] is not disconnected it is connected[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].

Notes

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

References

  1. 1.0 1.1 1.2 1.3 1.4 1.5 Introduction to Topological Manifolds - John M. Lee