Discussion
Let X be a locally connected topological space. Then which of the following is true ?
Answer: C
The true statement is every component of an open set of X is open in X.
This is a standard characterisation: X is locally connected if and only if the components of every open subset are open. If U is open and C is a component of U, each point of C has a connected open neighbourhood inside U, which must lie in C, so C is open.
Local connectedness does not imply connectedness (two disjoint intervals are locally connected but not connected). Nor does it give local path connectedness or make components path components: Kirch's topology on the positive integers is a countable, connected, locally connected Hausdorff space, and a countable Hausdorff space has no non-constant paths, so its single component is not path connected.
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