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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 (?))
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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.


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):