Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [locally-connected]

The tag has no usage guidance.

1
vote
0answers
21 views

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-...
1
vote
1answer
67 views

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 ...
3
votes
1answer
50 views

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(...
0
votes
0answers
62 views

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 ...
1
vote
1answer
30 views

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 $0 ≤ y ≤ 1$ $...
2
votes
1answer
107 views

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 ...
0
votes
1answer
48 views

About local linear connect [closed]

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 ...
0
votes
1answer
33 views

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" ...
0
votes
0answers
106 views

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 \...
2
votes
2answers
123 views

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 $...
2
votes
1answer
54 views

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 ...
2
votes
1answer
24 views

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 ...
0
votes
2answers
205 views

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 ...
0
votes
1answer
130 views

$ \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} $ ...
2
votes
1answer
188 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 $[0,1]$...
1
vote
4answers
713 views

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 ...
0
votes
2answers
146 views

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 $B_r(x)$ ...
2
votes
2answers
74 views

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 ...
1
vote
0answers
96 views

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 ...
8
votes
1answer
175 views

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 $[0,1]$, ...
2
votes
0answers
35 views

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 ...
2
votes
1answer
82 views

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 ...
1
vote
2answers
69 views

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 ...
6
votes
3answers
126 views

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 $S$ ...
1
vote
1answer
140 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 ...