Questions tagged [locally-connected]

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.

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Number of components of an open subspace of a compact locally connected space

In a locally connected space $X$, one can show that the connected components of any open subspace $U\subseteq X$ are all open in $X$ (cf. Theorem 25.3 in Munkres' Topology 2e). Therefore, if $X$ is ...
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Is an open connected subspace of $\mathbb{R^2}$ locally path-connected?

I want to proof the following statement : Let X be an open connected subspace of $\mathbb{R^2}$. Show that X is also path connected. The standard way to prove this problem from what I at least saw was ...
muhammed gunes's user avatar
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Adding point to connected open set

Let $X$ be a compact, connected, locally connected space. Let $U$ be a connected open subset of $X$. Let $p\in \overline U$. Clearly $U\cup \{p\}$ is connected. Is $U\cup \{p\}$ locally connected? Is $...
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Metric space that can be written as the finite union of connected subsets but isn't locally connected

I'm looking for an example of a metric space $X$ such that for every $\epsilon > 0$ there exist connected subsets $A_1, \dots A_n$ for some $n \in \mathbb{N}$ such that $X = \cup_{i = 1}^nA_i$ and ...
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Definition of local connectedness by Zorich's Mathematical Analysis II

I am reading chapter 9,section 4,of Mathematical Analysis II, written by Zorich. In Exercise 4 of the fourth section he defined the locally connectedness: a topological space $\left(X,\tau\right)$ is ...
MGIO's user avatar
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14 votes
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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 ...
Jakobian's user avatar
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Proof about locally connected

I have a doubt with my proof of the following result: A topological space $(X,\mathcal{F})$ is locally connected if and only if for every open set $U \subset X$ each connected component of $U$ is open ...
LH8's user avatar
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Prove that product of two locally connected spaces is locally connected. [duplicate]

Let $X$ and $Y$ be two locally connected spaces. I need to show that $Z = X \times Y$ is locally connected. Here is my attempt: Proof. Let $z=(x, y)\in Z$ and $N$ be any neighborhood of $z$. I need to ...
Chingis's user avatar
3 votes
2 answers
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Is pointwise locally connectedness preserved under a quotient map?

Let $(X, \mathcal{T}_X)$ and $(Y, \mathcal{T}_Y)$ be topological spaces. $X$ is locally connected at $x \in X$, if for each $U \in \mathcal{T}_X(x)$ there exists $U' \in \mathcal{T}_X(x)$ such that $U'...
kaba's user avatar
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Quotient of R by irrationals with same absolute value

while studying some general topology I came across the following topological space: on $\mathbb{R}$ we define an equivalence relation by $x \sim y$ if and only if $x=y$ or $|x|=|y|$ and $x \notin \...
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Potential errata found: show the non locally connected points of a metric space form a discrete space, under a given new metric

I am studying Topology and Groupoids by Ronald Brown, 3rd Edition. I believe I have spotted an errata in a given question, as I seem to have found a counterexample to what it wishes me to prove. I ...
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Consider the space $(\Bbb R, \tau_l)$, where $\tau_l$ is the lower limit topology. Is this space locally connected?

Consider the space $(\Bbb R, \tau_l)$, where $\tau_l$ is the lower limit topology. Is this space locally connected? I first thought that this would be true since $\tau_l$ is finer than the standard ...
Walker's user avatar
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Let $X$ be a locally connected space and $f:X \to Y$ a continuous closed surjection. Show that $Y$ is locally connected.

Let $X$ be a locally connected space and $f:X \to Y$ a continuous closed surjection. Show that $Y$ is locally connected. Let $V \subset Y$ be an open set and $C$ a component containing $V$. To prove ...
Walker's user avatar
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Let $X=\{(x,y) \in \Bbb R^2 \mid x = 0 \text{ or } x^2+y^2 \in \Bbb Q\} \subset \Bbb R^2$. Is $X$ connected? Is it locally-connected?

Let $X=\{(x,y) \in \Bbb R^2 \mid x = 0 \text{ or } x^2+y^2 \in \Bbb Q\} \subset \Bbb R^2$. Is $X$ connected? Is it locally-connected? $X$ is the $y$-axis union rational points on the unit disc. I ...
Alucard's user avatar
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neighbourhood open basis for a connected set in $\mathbb C$

Given $X$ a connected and locally connected space, let $S\subseteq X$ a connected set. Given any open $U$ containing $S$, is it true that there exists an open and connected $V$ such that $S\subseteq V ...
Exodd's user avatar
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Is arbitrary intersection of locally connected topologies locally connected?

Let $X$ be a set and $\{\mathcal{T}_i \subset \mathcal{P}(X) : i \in I\}$ be an arbitrary set of locally connected topologies in $X$; it may or may not contain all of them and $I$ may be finite or ...
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Is intersection of refining locally connected topologies locally connected?

Let $(X, \mathcal{T})$ be a topological space, $S(X)$ be the set of topologies in $X$, and $$\mathcal{T}^P = \bigcap \{\mathcal{T}^* \in S(X) : \mathcal{T} \subset \mathcal{T}^* \textrm{ and } (X, \...
kaba's user avatar
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A weakly locally path connected space that is not locally path connected

I am reading through an article by Shelah where it has a definition for a weakly locally path connected space: Say that $X$ is weakly locally path connected (WLPC) if for every $x\in X$ and every ...
Kooranifar's user avatar
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Restriction of Covering Spaces to a component.

Let $p:X\rightarrow Y$ be a covering space (i.e. locally over $Y$, $p$ looks like the projection $U\times F\rightarrow U$, with $F$ discrete). If $Y$ is locally connected, it is known that the ...
Leopoldo's user avatar
3 votes
4 answers
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Connected set which is no-where path connected

Background: It's a fun exercise to try to construct a connected space $T$ such that no two points in $T$ can be connected with a path. My solution to the puzzle was to use an order topology on a ...
Dark Malthorp's user avatar
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Counterexample of a disconnected topological space with open components but not locally connected

Theorem 25.3 in Munkres proved that a space is locally connected if the components of every open subset is open. What obviously follows from the theorem is that the components of a locally connected ...
cicolus's user avatar
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Intuitively, what is the difference between a "simply connected" set and a "locally connected" set?

I was looking into the Mandelbrot Set and saw a note that said it has been proven that the Mandelbrot Set is "simply connected" but it is still an open question of whether or not it is "...
e4494s's user avatar
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Characterizing locally connected continua

I am looking for a reference to the proof of the following result: Let $E\subseteq\mathbb C$ be a continuum. Then the following assertions are equivalent: (1) $E$ is locally connected; (2) (i) each ...
ray's user avatar
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Show that the n-sphere $\mathbb S^{n}$ is locally connected

A topological space $(X,\tau)$ is locally connected iff $\forall x\in X$ ,$\exists \{v_{i} : i\in I\}$ a familly of connected neighberhoods of $x$ such that $\forall u$ neighberhood of $x$ $\exists i\...
Abdellatif Ouhaddou's user avatar
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Local connectedness is a topological property [duplicate]

I am trying to prove that local connectedness is a topological property. Then what do I need to show? Do I need to show that if $X$ and $Y$ are homeomorphic topological spaces (let $f$ be the ...
Esha's user avatar
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Locally connected for sub space topology of $\Bbb R^2$

Consider the following subspace $X:=\Bbb R^2\setminus(\Bbb Q\times \Bbb Q)$ of $\Bbb R^2$, where $\Bbb R^2$ with the usual topology. I would like to check this space in the terms of various kinds of ...
Gob's user avatar
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Proving/disproving a claim on the relationship between the kinds of connectedness (the usual, and path and local)

Context: I am given, as an exercise, the task of proving this claim: For $X$ a topological space, show it is path-connected if and only if $X$ is connected and each $x \in X$ lies in some path-...
PrincessEev's user avatar
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A compact set $X$ is only locally conneted in on point but not locally connected in other points in $X$

When study the theorem of boundary correspondence in complex analysis, I come into connact with the concept of Locally Connectedness and several examples. Definition: A compact set $X\subset\mathbb{C}$...
user823011's user avatar
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2 answers
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topological properties of $X= \left( \cup_{n=1}^{\infty} \{\frac{1}{n}\} \times [0,1] \right) \cup (\{0\} \times [0,1]) \cup ( [0,1] \times \{0\})$

Let $X= \left( \cup_{n=1}^{\infty} \{\frac{1}{n}\} \times [0,1] \right) \cup (\{0\} \times [0,1]) \cup ( [0,1] \times \{0\}) \subset \mathbb{R}^2$. I want to show whether this space $X$ is normal and ...
phy_math's user avatar
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2 votes
1 answer
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Example of a map of coverings which is not a covering map, if the base space is not locally path-connected

Consider two coverings $X \to Y$ and $X^\prime \to Y$. A surjective map $X \to X^\prime$, which fits in a commutative triangle with the covering maps, is again a covering, given that $Y$ is locally ...
HDB's user avatar
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5 votes
1 answer
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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 ...
Pepe's user avatar
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Types of connectedness for $L_n:=\{(x,\frac{x}{n})\mid0\leq x\leq1\}\subset\mathbb{R}^2$ for $n\in\mathbb{Z}_+$

Exercise: Let $L_n:=\{(x,\frac{x}{n})\mid0\leq x\leq1\}\subset\mathbb{R}^2$ for $n\in\mathbb{Z}_+$. Then consider $X=\{(1,0)\}\cup\bigcup_{n=1}^\infty L_n\subset\mathbb{R}^2$ together with the ...
Laplace's Demon's user avatar
2 votes
0 answers
66 views

What is this topological property? A form of local connectedness?

Let $S$ be a subset of some topological space $X$. What is known about the following property of $S$? Does it have a name? Is it standard? Property: For all $x\in \bar{S}$ (the closure of $S$), and ...
cfp's user avatar
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3 votes
0 answers
110 views

A sheaf is locally constant iff the etale space is a covering

Let $X$ be a locally connected space a sheaf $F$ on $X$ is called locally constant whenever each $x\in X$ has a basis of open neighbourhoods$N_x$ and if $U,V \in N_x$ such that $U$ is a subset of $V$ ...
user850424's user avatar
4 votes
2 answers
386 views

Is every open and connected set in $\mathbb C$ the continuous image of the open unit disk?

Let $$ \mathbb D=\{z\in\mathbb C\ |\ |z|<1\} $$ be the open unit disk in $\mathbb C$. It is well known that an open (nonempty) set $U\subseteq\mathbb C$ is simply connected if and only if it is ...
sranthrop's user avatar
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Is the local connectedness heritable over the connected subspaces?

Definition A topological space $X$ is locally connected if each point $x\in X$ has a base of connected neighborhoods. So let be $X$ a locally connected space and let be $Y$ a connected subspace. So if ...
Antonio Maria Di Mauro's user avatar
2 votes
1 answer
98 views

Boundaries in Spaces where Quasicomponents and Components Coincide

Let's call a space $X$ geometric if its components and quasi-components coincide. Let's also define a property called the boundary bumping property: $X$ has the boundary bumping property ("bbp&...
John Samples's user avatar
1 vote
1 answer
104 views

X has finitely many connected components, can we get X locally connected?

say X has finitely many connected components, can we get X locally connected? I'm thinking to prove this and then I can have every component is open. Thanks!
angushushu's user avatar
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1 answer
90 views

Proof Verification : Equivalent Condition for Locally Connected Space

A topological space $X$ is locally connected if for every $x$ in $X$ and for every open set $V$ containing $x$, there is a connected open set $U$ with $x \in U \subset V$. I think it is equivalent ...
with-forest's user avatar
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1 answer
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Local connectedness of topologist's comb

I'm using Croom's textbook, and came with the following definition of local connectedness: A space $X$ is locally connected at a point $p$ in $X$ if every open set containing $p$ contains a connected ...
WishofStar's user avatar
2 votes
0 answers
48 views

Metric space locally compact but not uniformly locally connected

Definition: A set $M$, also in a metric space, is said to be uniformly locally connected if and only if for every $\varepsilon > 0$ there exists $\delta>0$ such that any pair of points $x, y$ of ...
user123456's user avatar
1 vote
0 answers
37 views

Show that if $ X $ has the property $ \mathcal{M} $, then $ X $ is locally connected.

Let $ (X, \tau) $ be a topological space. We will say that $ X $ has the property $ \mathcal{M} $ if every open coverage of $ X $ admits a finite refinement consisting of connected sets. Show that if $...
mathusu16's user avatar
1 vote
1 answer
75 views

If every open cover of $ X $ admits a finite refinement consisting of connected sets, then $ X $ is locally connected.

Let $ (X, \tau) $ be a topological space. If every open coverage of $ X $ admits a finite refinement consisting of connected sets, then $ X $ is locally connected. Proof: let $\{U_i\} $a open coverage ...
mathMaría's user avatar
1 vote
0 answers
105 views

$X$ a compact Hausdorff space has the property $ \mathcal{M} $ if, and only if $X$ is locally connected.

$(X,\tau)$ a topological space. $X$ a compact Hausdorff space has the property $ \mathcal{M} $ if, and only if $X$ is locally connected. Having the $ \mathcal{M} $ property means: Let $ (X, \tau) $ ...
Us12's user avatar
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2 votes
1 answer
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X is connected and locally connected

Let $X$ be a $T_1$ topological space. Show that $X$ is connected and locally connected if and only if for each open cover $\{U_s\}_{s\in S}$ and any $x, y \in X$, there exist $s_1,. . . , s_n \in S$ ...
user08's user avatar
  • 251
5 votes
1 answer
93 views

$X$ 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 ...
user1999's user avatar
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the boundary of a component $C$ of $A$ is contained in the boundary of $A $.

Let $(Y, \tau)$ be a locally connected topological space and $A\subset Y$. Let $C$ be a component of $A$. Show that $\partial C\subset \partial A$. I proved that $Int_{Y}(C)= C\cap Int_{Y}(A)$. Using ...
User1997's user avatar
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10 votes
1 answer
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$ 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. ...
user08's user avatar
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0 answers
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Possibly differing definitions of local path-connectedness

In C.C.Pugh's book "Real Mathematical Analysis" a metric space $X$ is said to be locally path-connected if for each $p\in X$ and (open) neighborhood $U$ containing $p$ there is a path-...
SyntasticMonoid's user avatar
1 vote
2 answers
187 views

Closure of topology sine $Y = \lbrace (x,\sin(\frac{1}{x})) : x > 0 \rbrace $ [duplicate]

Given the subspace $Y = \lbrace (x,\sin(\frac{1}{x})) : x > 0 \rbrace \subset \mathbb{R}^2$, I know that $Y$ is connected being the image of a connected of $\mathbb{R}$ (i.e. $(0,+\infty)$.), ...
jacopoburelli's user avatar