proof that components of open sets in a locally connected space are open

Theorem.

A topological space $X$ is locally connected if and only if each component of an open set is open.

Proof.

First, suppose that $X$ is locally connected and that $U$ is an open set of $X$. Let $p\in C$, where $C$ is a component of $U$. Since $X$ is locally connected there is an open connected set, say $V$ with $p\in V\subset U$. Since $C$ is a component of $U$ it must be that $V\subset C$. Hence, $C$ is open. For the converse, suppose that each component of each open set is open. Let $p\in X$. Let $U$ be an open set containing $p$. Let $C$ be the component of $U$ which contains $p$. Then $C$ is open and connected, so $X$ is locally connected.

∎

As a corollary, we have that the components of a locally connected space are both open and closed.

Title	proof that components of open sets in a locally connected space are open
Canonical name	ProofThatComponentsOfOpenSetsInALocallyConnectedSpaceAreOpen
Date of creation	2013-03-22 17:06:07
Last modified on	2013-03-22 17:06:07
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	6
Author	Mathprof (13753)
Entry type	Theorem
Classification	msc 54A99