# Questions tagged [locally-connected]

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### Characteristic of a connected, locally compact, Hausdorff space X which is locally connected.

Q. Prove that a connected, locally compact, Hausdorff space X is locally connected if and only if for each compact subset K and each open set U containing K, all but a finite number of components of X-...
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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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### If G/H is connected and H is connected then is true that G is connected? [duplicate]

Let G be a topological group. If G/H and H are connected then is true that G is connected?
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### Locally connected and connected

Let X be locally connected, $A \subset X$ arbitrary. Let $S \subset A$ be connected and open in A. Show $S=U\cap A$ where $U$ is connected and open. I think that I have solved already but I never ...
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### Is connected component open?

There is a theorem that:A space is locally connected iff each connected components of an open set is open. But recently I had seen to prove That each connected component is closed. Connected ...
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### The subgraph induced by the neighbourhood of a vertex

I have the connected graph as in picture 1 attached. Is it correct if I take the subgraph induced by the neighbourhood of the vertex "a" as in the picture 2 below? The neighbouring vertices of "a" ...
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### Not locally connected but path connected particular example

Show an example of a path connected topological space that it's not locally connected. Example (by William Eliot): In $\mathbb R^2$, draw lines from $(0,0)$ to $(1,q)$ for all rational $q$ in $[0,1]$...
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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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### Connected neibourhood and path connected neibourhood

I studying locally connectedness and locally path connectedness from MUNKRES but I can't understand the term connected neibourhood and path connected neibourhood. I will be thankful ,if anyone ...
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### Set of All Points at which $X$ is Locally Connected

The following question came to me as I was reading about local connectedness: Let $$S := \{x \in X \mid X \text{ is locally connected at } x \},$$ where $X$ is some topological space. Is $S$ ...
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### Example of a topological space that is connected, not locally connected and not path connected

I know that "The topologist's sine curve" is an example of a topological space that is connected, not locally connected and not path connected. https://en.wikipedia.org/wiki/Topologist%27s_sine_curve ...
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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 $\prod_{j \in J} X_{j}$ ...
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### Characterisation of One Dimensional Peano Continua

Question: Is every one dimensional peano continuum a countable union of arcs which pairwise intersect only in their end points? Definitions: Continuum - A connected, compact metric space Peano ...
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### Any example of a connected space that is not locally connected?

A topological space $X$ is said to be locally connected at a point $x \in X$ if, for every neighborhood $U$ of $x$ (i.e. open set $U$ such that $x \in U$), there exists a connected neighborhood $V$ of ...
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### Locally connected implies disjoint union of open connected sets

I'm struggling with the following question (from an introductory analysis course): A metric space $E$ is said to be locally connected if for all $x \in E$, there exists $r > 0$ such that $B_r(x)$ ...
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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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### The Hahn-Mazurkiewicz Theorem for non-metric spaces

I am looking for a yes or no answer to the following question, though if there is a simple explanation I'd like that as well: If $X$ is any continuous image (not necessarily metrizable) of $[0,1]$, ...
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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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### 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 ...
I am stuck on a problem! Suppose $X$ is a compact, connected metric space. Let $L(X)$ be the set of points at which $X$ is not locally connected (here, locally connected means the point has a local ...
Is the property of local connectedness or local path-connectedness invariant under homotopy equivalence? I.e. If $X,Y$ are homotopically equivalent Hausdorff topological spaces such that $X$ is ...