Questions tagged [locally-connected]

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Local connectedness is preserved under retractions

I want to show that if $X$ is a locally connected topological space, $A\subseteq X$ is a subspace and $f:X \rightarrow A$ is continuous such that $f|_{A} = Id_{A}$, then $A$ must be locally connected ...
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Is there a name for local connectedness except you use punctured neigborhoods instead?

Take the definition for a locally connected space: A space is locally connected at $c$ if for every neighborhood of $c$ contains an connected open neighborhood of $c$. A space is loccally connected ...
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The idea of weakly locally connected (connected im kleinen) points, and counterexamples.

I've recently had the chance to learn about the weak local connectedness (connected im kleinen) property, of points in topological spaces. Also, I've been trying to construct a space which has some ...
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Hausdorff, locally connected and locally compact space reference.

I would like to find a reference to the following proposition: Let $X$ a Hausdorff, locally compact, locally connected, connected space and $K \subseteq X$ a compact subset. Then, there exists a ...
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Existence of Connected Sets in Complements of Closed Sets

Suppose that we have a connected $T_1$ space $X$ and two disjoint, closed subsets $A, B$. Then if $X$ is also locally connected, it is a theorem that there is a connected subset $C \subset X \...
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$ \prod_{j \in J} X_{j} $ is locally connected if, and only if, each $ X_{j} $ is locally connected …

I do not understand two parts of the converse of the following theorem: Let $ \lbrace X_{j}: j \in J \rbrace $ be a family of non-empty topological spaces. The product space $ \prod_{j \in J} X_{j} $ ...