For questions on locally connected topological spaces. A topological space is called locally connected if every neighborhood of every point contains a connected open neighborhood.
Questions tagged [locally-connected]
104 questions
76
votes
3 answers
Path-connected and locally connected space that is not locally path-connected
I'm trying to classify the various topological concepts about connectedness. According to 3 assertions ((Locally) path-connectedness implies (locally) connectedness. Connectedness together with locally path-connectedness implies…
qyy
- 1,187
15
votes
2 answers
Does locally compact, locally connected, connected metrizable space admit a metric with connected balls?
If $X$ is a locally compact, locally connected, connected metrizable space, does that imply that there must be a metric $d$ on $X$ such that $B(x, r) = \{y\in X : d(x, y) < r\}$ is connected for all $x\in X, r > 0$ ? Note that if $X$ admits such…
Jakobian
- 16,581
11
votes
4 answers
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 $x$ such that $V \subset U$. If $X$ is locally…
Saaqib Mahmood
- 27,852
10
votes
1 answer
Hereditarily locally connected spaces
Here, a space is called hereditarily locally connected ("hlc"), if each subset is locally connected. Note that this is much stronger than the usual definition, which requires only connected subspaces to be locally connected. For instance, $\mathbb…
Ulli
- 7,406
10
votes
1 answer
$ X = A \cup B $ where $ A $ and $ B $ are closed and $ A \cap B $ is locally connected. Show that $A$ and $B$ are locally connected.
Let $(Y, \tau)$ a locally connected topological space. suppose $ Y = A \cup B $ where $ A $ and $ B $ are closed and $ A \cap B $ is locally connected. Show that $A$ and $B$ are locally connected. let's see that A is locally connected. Let $x \in A$…
user08
- 251
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10
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3 answers
Properties of space $X = T \cup \bigcup_{n=1}^{\infty} S_n \cup \{ (0,q) : q \in \mathbb Q \wedge q \in [0,1] \} $
Let $S_n$ be closed interval in euclidean plane $\mathbb R^2$ with ends in $(\frac{-1}{n}, \frac{1}{n})$ and $(\frac{-1}{n}, 1)$ . Moreover let $T$ be boundary of triangle with vertices $(1,1),(0,0),(-1,1)$ . A. Prove that subspace of plane $$ X = T…
newuser458
- 427
8
votes
0 answers
Simply connected neighbourhood of a simply connected, locally path connected, closed set
This is a follow up of this question. Let $M$ be a smooth manifold and let $C \subseteq M$ be a closed, simply connected and locally path connected subset. Is it possible to find an open, simply connected, neigbourhood of $C$ ?
Math_tourist
- 439
8
votes
1 answer
The Hahn-Mazurkiewicz Theorem for non-metric spaces
I am looking for a yes or no answer to the following question, though if there is a simple explanation I'd like that as well: If $X$ is any continuous image (not necessarily metrizable) of $[0,1]$, then is $X$ necessarily locally connected? Just…
Forever Mozart
- 9,594
7
votes
2 answers
Countable basis but uncountably many connected components
Looking for some guidance on two topology questions: (a) Show that a locally connected space with a countable basis, has at most countably many connected components. (b) Give an example when X has countable basis but it has uncountable many…
PistolsAtDawn
- 93
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7
votes
3 answers
Set of All Points at which $X$ is Locally Connected
The following question came to me as I was reading about local connectedness: Let $$S := \{x \in X \mid X \text{ is locally connected at } x \},$$ where $X$ is some topological space. Is $S$ generally open in $X$? If not, suppose it is. Would this…
user193319
- 8,490
6
votes
1 answer
Relation between free arcs and branch points in continua
before I begin, I would like to provide some definitions and theorems. Definition. A topological space $(X, \tau_X)$ is a continuum , if $X$ is a non-empty, metric, compact and connected space. Definition. Let $X$ be a continuum and define $E(X)=\{p…
Aldo
- 179
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- 14
5
votes
1 answer
A result about trees in topology.
before I begin, I would like to provide some definitions and theorems. Definition. A topological space $(X, \tau_X)$ is a continuum , if $X$ is a non-empty, metric, compact and connected space. Definition. Let $X$ be a continuum and define $E(X)=\{p…
Aldo
- 179
- 1
- 14
5
votes
1 answer
Locally connectedness preserved after removing one point
Given a locally connected space $X$ with two or more points is it always true that for any point $x$ in $X$ the subspace $X \setminus \{x\}$ is also locally connected? I have proved it for Hausdorff spaces but I would like to know if it is true for…
Pepe
- 128
5
votes
1 answer
$X$ is locally connected and countably compact
Let $(X,\tau)$ be a topological space $T_3$ . Show that the following statements are equivalent: Every open and finite coverage of X has a finite refinement consisting of connected sets. Space X is locally connected and countably compact. A…
user1999
- 504
5
votes
1 answer
Why is there no "weakly version" of locally path-connectedness?
As usual in topology, there are many different definition for terms like "locally connectedness": Let $(X,\mathcal{T})$ be a topological space. Note that I use the following definition of neighbourhood: A set $U\subset X$ is a neighbourhood of $x\in…
user674359