Connected not implies locally connected

This article gives the statement and possibly, proof, of a non-implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property (i.e., Connected space (?)) need not satisfy the second topological space property (i.e., Locally connected space (?))
View a complete list of topological space property non-implications | View a complete list of topological space property implications |Get help on looking up topological space property implications/non-implications
Get more facts about connected space|Get more facts about locally connected space

Contents

Statement

It is possible for a topological space to be a connected space but not a locally connected space.

Definitions used

Connected space

Further information: connected space

A topological space is termed connected if it cannot be expressed as a disjoint union of two nonempty open subsets.

Locally connected space

Further information: Locally connected space

A topological space $X$ is termed locally connected if, for every point $x \in X$ and every open subset $U$ of $X$ containing $x$, there exists an open subset $V$ of $X$ such that $x \in V$, $\overline{V}\subseteq U$, and $V$ is a connected space.

Proof

The general idea behind counterexamples is that the connecting apparatus between a point and points very close to it is via points that are very far from it. Most of these counterexamples are also counterexamples for the related fact that path-connected not implies locally path-connected, where points close by can be connected only via paths that go through points that are far away.

Here are some counterexample spaces (more elaboration needed):

• The infinite broom and closed infinite broom. Both of these are connected spaces, and the latter is also a path-connected space. However, neither of these is locally connected.
• The topologist's sine curve is a connected space but not a locally connected space.
• The comb space is a connected space -- in fact, it is a contractible space. However, it is not locally connected.