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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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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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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Why is Cantor space not locally connected? Confused about locally connected sites.

Let $C$ be a category endowed with a Grothendieck topology $J$ of covering sieves, giving us a site $(C,J)$. We say $(C,J)$ is locally connected if for any object $c\in C$ all covering sieves $j\in J(...
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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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Example of a Connected Space which is not Locally Connected.

Here is an good example of a connected space which is not locally connected but I am stuck in disproving the local disconnectedness: Let $A$ $=$ $\big\{ (x, y)| x \in \mathbb{Q}^c $ and $0 ≤ y ≤ 1$ $...
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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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191 views

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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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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Show that $CX$ is locally (path-)connected iff $X$ is

I'm doing some excercise in Lee's book. I've come up with a solution but not really sure. I also want to see other people solutions too. Here is the problem: Let $X$ be any topological space and $...
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1answer
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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 ...
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Set of Points where X Fails to be Locally Connected

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 ...
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150 views

Is the property of local connectedness or local path-connectedness invariant under homotopy equivalence?

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 ...