Questions tagged [locally-connected]

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About locally path-connected spaces

Let $X$ be a locally path-connected space such that every point $x$ has a neighbourhood $U$ with $\overline{U}$ compact. I need to show that the path-components of $X$ are open and are the same as the ...
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Are components of complement of a set $S$ always close to $S$

Suppose that $X$ is a connected topological space with a connected subset $S$, and let $K$ denote a component of $X\setminus S$. (All sets listed are assumed non-empty, apart from $\emptyset$ of ...