Connected not implies locally connected

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

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
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 particular example::infinite broom and particular example::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 particular example::topologist's sine curve is a connected space but not a locally connected space.
 * The particular example::comb space is a connected space -- in fact, it is a contractible space. However, it is not locally connected.