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]
99 questions
74
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…
Qian
- 1,167
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
- 15,280
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
- 27,542
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…
user08
- 251
10
votes
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
$$…
newuser458
- 417
8
votes
0 answers
Simply connected neighbourhood of a simply connected, locally path connected, closed set
This is a follow up of this question. Let $M$ be a smooth manifold and let $C \subseteq M$ be a closed, simply connected and locally path connected subset.
Is it possible to find an open, simply connected, neigbourhood of $C$?
Math_tourist
- 439
8
votes
1 answer
The Hahn-Mazurkiewicz Theorem for non-metric spaces
I am looking for a yes or no answer to the following question, though if there is a simple explanation I'd like that as well:
If $X$ is any continuous image (not necessarily metrizable) of $[0,1]$, then is $X$ necessarily locally connected?
Just…
Forever Mozart
- 9,534
7
votes
2 answers
Countable basis but uncountably many connected components
Looking for some guidance on two topology questions:
(a) Show that a locally connected space with a countable basis, has at most
countably many connected components.
(b) Give an example when X has countable basis but it has uncountable…
PistolsAtDawn
- 93
- 5
7
votes
1 answer
A result about trees in topology.
before I begin, I would like to provide some definitions and theorems.
Definition. A topological space $(X, \tau_X)$ is a continuum, if $X$ is a non-empty, metric, compact and connected space.
Definition. Let $X$ be a continuum and define $E(X)=\{p…
Aldo
- 293
- 5
7
votes
3 answers
Set of All Points at which $X$ is Locally Connected
The following question came to me as I was reading about local connectedness:
Let $$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
- 8,318
6
votes
1 answer
Free arcs in the Universal dendrite
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 $E(X)=\{p \in X: ord_X(p)=1\}$, $O(X)=\{p \in X: ord_X(p)=2\}$ and $R(X)=\{p \in…
Aldo
- 293
- 5
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…
user674359
5
votes
1 answer
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 locally connected / locally path connected, then is it…
user228169