Use this tag for question on connected spaces and various related notions (connected components, locally connected spaces, pathwise/arcwise connected spaces, simply connected spaces, totally disconnected spaces ...) and for connectedness in graph theory.

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2
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0answers
23 views

When does “every closed path is homotopic to a point” imply the space is path connected?

In the middle of talking about primitives and the Cauchy integral theorem, my Complex Analysis teacher came up with this sentence: This reasoning can be done in any simply connected set, because ...
0
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1answer
20 views

Counting weakly connected graphs with outdegree of exactly one.

If we count all graphs of $N$ labelled vertices, where each vertex has an outdegree of exactly $1$ with no self-loops allowed, we'll find that there are exactly $(N-1)^N$ of them (for every of $N$ ...
0
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1answer
15 views

For every $n \ge 2 $ , existence of uncountably many mutually disjoint closed balls , complement of whose union is path connected

For every $n \ge 2 $ , does there exist an uncountable family $\{D_\lambda: \lambda \in \Gamma\}$ of mutually disjoint closed balls in $\mathbb R^n$ , such that $\mathbb R^n \setminus \cup_{\lambda ...
1
vote
1answer
25 views

Complement of the union of finitely many , mutually disjoint , closed balls in $\mathbb R^n$ , for $n>1$ , path connected? [on hold]

Let $D_1,D_2,...,D_k$ be finitely many , mutually disjoint , closed balls in $\mathbb R^n$ , where $ n \ge 2$ . Then is $\mathbb R^n \setminus \cup_{i=1}^k D_i$ path connected ?
0
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0answers
34 views

When is a connected space, path connected? [on hold]

Let $X$ be a connected topological space. When is $X$ path connected? Is the Hausdorff property enough? Is it too much?
2
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1answer
39 views

Is this argument invalid?

So, what I was trying to prove is: Let $\pi : X \to Y$ the quotient map such that $\pi^{-1}(\{y\})$ connected for all $y \in Y$. Then $X$ is connected. Suppose yet that $X$ is locally connected. ...
1
vote
3answers
148 views

Show the following set is connected

For any $x \in \Bbb R^n$ how do I show that the set $B_x := \{{kx\mid k \in \Bbb R}$} is connected. It should also be concluded that $\Bbb R^n$ is connected. I was thinking of starting by assuming ...
0
votes
1answer
37 views

removing rationals in topolosist's sine curve

Munkres pg. 160 says, where $S$ is the topologist's sine curve If one forms a space from $\bar S$ by deleting all points of $V$ having rational second coordinate, one obtains a space that has only ...
2
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0answers
33 views

Other counterexamples for this problem on connectedness

If $A$ is a connected subspace of $X$, does it follow that $\operatorname{Int}A$ and $\operatorname{Bd}A$ are connected? Does the converse hold? Justify your answers. For original problem, I can ...
0
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2answers
23 views

Is this subset of the plane path connected?

Inspired by this question I asked myself the question which I am going to describe: Let $\mathbb {I}$ be the set of all irrational numbers. Let $\mathbb {I}^2$ be the Cartesian product of the set ...
4
votes
0answers
41 views

Show that $X-C$ is connected [duplicate]

Let $X$ be a connected metric space, let $A \subseteq X$ be a connected set. Let $\mathcal{C}$ be a connected component of $X-A$. Show that $X-\mathcal{C}$ is connected. Ok, so Im been dealing ...
0
votes
1answer
63 views

How can I prove that a set is connected?

For example,if $A \subseteq \mathbb{R}^2$ is finite, so $\mathbb{R}^2 \backslash A$ is connected. I'm trying to use the negation of the definition of connected metric space , so can I reach a ...
3
votes
1answer
50 views

Connectedness of the Hausdorff distance.

Does anyone know a proof of connectedness of the Hausdorff distance? I mean a proof of the following: Theorem If $(X, \rho )$ is a connected metric space, then $(F(X), d_h )$ is also connected. ...
1
vote
0answers
24 views

Properties of Connected Closed Trails?

Consider the (sub)graph below. I am allowing for the possibility of additional edges and nodes, and just want to look at the features of this subgraph. What seems apparent is that there is not a ...
1
vote
2answers
50 views

Proposed proof of topological result

Hi I just want to find out if the following is a acceptable proof for the proposition: "Consider metric space $(X,d)$. If $A \subset X$ is connected and $A \subset B \subset \bar{A}$ then $B$ is ...
3
votes
2answers
76 views

Determining graph biconnection from degree sequence

The title is self-explanatory: having the degree sequence of a graph, how can I find whether it is biconnected? The fact that I can't manage to draw one with such property does not mean that it does ...
3
votes
1answer
70 views

If $X$ is not connected, then $X\subset G_1\cup G_2$

The following proposition is true? Let $X\subset\mathbb{R}^n$ be a not connected set. Then there are disjoint open sets $G_1,G_2\subset \mathbb{R}^n$ such that $X\subset G_1\cup G_2$ with $X\cap ...
2
votes
1answer
25 views

Does there exist a connected metric space, with more than one point and without any isolated point, in which at least one open ball is countable?

Does there exist a connected metric space, with more than one point and without any isolated point, in which at least one open ball is countable?
2
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0answers
41 views

Efficient algorithm for visiting all connected subcomponents of a connected graph

Starting from a graph, how to identify all connected subsets? Connected components, i.e., maximal connected subsets, can be found in linear time. For simple graphs, e.g., chains, $1-2-...-N,$ ...
2
votes
1answer
31 views

Path components of quotient space

Let $X$ be a topological space and $A \subseteq X$ a subspace. Is it true that $X/A$ is path-connected if and only if $A$ meets every path component of $X$? Intuitively this seems reasonable but I'm ...
5
votes
2answers
107 views

Suppose $S$ is bounded star shaped domain of $ \mathbb R^n$ where $n \ge 2$ then $ \partial S$ is connected?

Def: A set $S$ in the Euclidean space $ \mathbb R^n $ is called a star shaped domain if there exists $x_o$ in $S$ such that for all $x$ in $S$ the line segment from $x_0$ to $x$ is in $S$. From my ...
4
votes
2answers
88 views

Suppose $S$ is bounded star shaped subset of $ \mathbb R^n$ where $n \ge 2$ then $ \partial S$ is path connected

Def: A set $S$ in the Euclidean space $ \mathbb R^n $ is called a star shaped domain if there exists $x_o$ in $S$ such that for all $x$ in $S$ the line segment from $x_0$ to $x$ is in $S$. Some days ...
3
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4answers
92 views

Does there exist a path connected metric space , in which at least one open ball is countable ?

Does there exist a path connected metric space with more than one point , in which at least one open ball is countable ?
0
votes
2answers
60 views

$\mathbb{I}\times\mathbb{Q}\cup\mathbb{Q}\times\mathbb{I}$ and connectedness [closed]

Is $\mathbb{I}\times\mathbb{Q}\cup\mathbb{Q}\times\mathbb{I}$ connected, where $\mathbb{I}$ is the set of irrational numbers and $\mathbb{I}\times\mathbb{Q}\cup\mathbb{Q}\times\mathbb{I}$ is a ...
2
votes
1answer
43 views

Questions on the counterexample for “connectedness doesn't imply path-connectedness”

I am trying to understand the disproof/counterexample that 'a topological space that is connected may not be path connected'. Here is the explanation from "C Adams and R Franzosa - Introduction to ...
3
votes
1answer
57 views

Need proof/counterexample for boundary of star shaped domain

A set $S$ in the Euclidean space $ \mathbb R^n $ is called a star shaped domain if there exists $x_o$ in $S$ such that for all $x$ in $S$ the line segment from $x_0$ to $x$ is in $S$. Is the ...
2
votes
2answers
23 views

An arbitrary product of connected spaces is connected

Let $\lbrace X_\alpha \rbrace_{\alpha \in J}$ be an indexed family of connected spaces; let $X$ be the product space $$X=\prod_{\alpha \in J} X_\alpha$$ Let a$=(a_\alpha)$ be a fixed point of ...
31
votes
6answers
515 views

Is the complement of countably many disjoint disks path connected?

Let $\{D_n\}_{n=1}^\infty$ be a family of pairwise disjoint closed disks in $\mathbb{R}^2$. Is the complement $$ \mathbb{R}^2 -\bigcup_{n=1}^\infty D_n $$ always path connected? Here ...
1
vote
1answer
23 views

For each $n \in \mathbb{N}$ consider $A_n = \{1/n\}\times [0,1]$ and let $X=\bigcup_{n \in \mathbb{N}}A_n \cup \{(0,0)(0,1)\}$

For each $n \in \mathbb{N}$ consider $A_n = \{1/n\}\times [0,1]$ and let $X=\bigcup_{n \in \mathbb{N}}A_n \cup \{(0,0)(0,1)\}$. Show that $\{(0,0)\}$ and $\{(0,1)\}$ are connected components. ...
12
votes
1answer
233 views

Let $D$ be a bounded domain (open connected) in $ \mathbb C$ and assume that complement of $D$ is connected.Then show that $\partial D$ is connected

I am trying to prove the following famous result in Point Set Topology. Let $D$ be a bounded domain (open connected) in $ \mathbb C$ and assume that complement of $D$ is connected. Then show that ...
3
votes
2answers
22 views

Locally connected metric spaces

I have the following definition of a locally connected metric space. Given $(X,d)$ a metric space, $x \in X$ and given $U \ni x$ a neighbourhood. Then there exist a connected neighbourhood $V$ susch ...
1
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2answers
67 views

Finite “snakes” in a connected space

Given an open cover $\{G_t\}_{t \in T}$ of connected space $X$ and two points $a, b \in X$ show that there exists finite sequence of indices $t_1, t_2, \dots, t_n$ (a snake) such that $G_{t_i} \cap ...
1
vote
2answers
40 views

Why $\hat{\mathbb{C}}\setminus K$ connected $\implies {\mathbb{C}}\setminus K$ connected? ($K $ compact)

Let $\hat{\mathbb{C}}=\mathbb{C}\cup \{\infty\}$ denote the extended complex plane, with the usual topology.That is $U$ such that $U$ is open in $\mathbb{C}$ and the neighbourhoods of $\{\infty\}$ ...
0
votes
2answers
45 views

Show that $\mathbb{R}$ and $\mathbb{R}^n$ are not homeomorphic if $n\geq 2$

Show that $\mathbb{R}$ and $\mathbb{R}^n$ are not homeomorphic if $n\geq 2$. I want to use a connection-type argument. I thought of giving the following proof; Suppose that there exist such a ...
-1
votes
2answers
25 views

If $C \subset X$ and $\mathcal{U} \subset X$ is open. Is $C \cap \mathcal{U}$ open in $C$?

In an exercise regarding connection, I came to the following problem, I am given $C \subset X$ and $\mathcal{U} \subset X$ is open (where $(X,d)$ is a metric space). And I could use that is $C \cap ...
0
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1answer
15 views

Conceptual question regarding conection in metric spaces.

I have to give an example of two sets $A,B \subset \mathbb{R}$ such that both are connected, but $A\cup B$ is not. So I thought of a trivial example $(0,1) \subset \mathbb{R}$, $(2,3) \subset ...
0
votes
1answer
25 views

Connectedness of a cartesian product

Let $X,Y$ be topological spaces. Show that the productspace $X\times Y$ is connected $\Leftrightarrow X,Y$ are connected. Could someone give me some pointers? I'm not looking for a full solution, ...
0
votes
2answers
47 views

if two space are homotopy equivalent and one is connected, prove that the other is connected as well

I've tried using the definition of homotopy equivalent spaces which states that X and Y are homotopy equivalent if: There are continuous functions $f:X \rightarrow Y,g:Y \rightarrow X$ such that $f ...
1
vote
3answers
33 views

How continuity of $f$ and path-connectedness of $g$ results in $f\circ g$ to be path-connected?

Theorem 6.29 (p.213) of Introduction to Topology: Pure and Applied by C Adams and R Franzosa says: Assume that $f : X \rightarrow Y$ is continuous and $X$ is path connected. Then $f (X)$ is a ...
2
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2answers
69 views

minimal embeddings of topological spaces into connected spaces

Defintions: Let $X$ be a topological space. 1) A connected space $Y$ is a minimal connected ambient (m.c.a for short) space for $X$ if there exists an embedding $i:X\mapsto Y$, and for every ...
2
votes
1answer
20 views

Proof product of components in factors is a component in product topology

Let $x = (x_1, x_2, .... x_{n})$ be a point in a product space $(Y, \tau_{Y}) = \prod_{i = 1}^{n} (X_{i}, \tau_{i})$. The component $C_{X}(y)$ in a topological space is the union of all connected ...
4
votes
2answers
209 views

Space which is neither locally connected at any point nor totally disconnected

Let $X$ be a topological space; then we say that $X$ is locally connected at $x$ if $x$ admits a neighborhood basis of open, connected sets. In this sense, a space is locally connected iff it is ...
0
votes
1answer
27 views

If the boundary of a convex set in $\mathbb R^n$ ($n>1$) is connected , is it necessarily also path-connected ?

If the boundary of a convex set in $\mathbb R^n$ ( where $n>1$) is connected , is it necessarily also path-connected ?
0
votes
1answer
18 views

Let $A$ be a convex set in $\mathbb R^n $ , where $n>1$ ; then is it true that $\bar A \setminus A^{\circ}$ i.e. the boundary of $A$ is connected ?

Let $A$ be a convex set in $\mathbb R^n $ , where $n>1$ ; then is it true that $\bar A \setminus A^{\circ}$ i.e. the boundary of $A$ is connected ? if yes , then is it also path connected ?
0
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1answer
41 views

Prob. 3 (b), Sec. 27 in Munkres' TOPOLOGY, 2nd ed: How does the $K$-topology on $\mathbb{R}$ differ from the usual topology?

Let $$ K \colon= \left\{\ \frac{1}{n} \ \colon \ n \in \mathbb{N} \ \right\},$$ and let the $K$-topology on $\mathbb{R}$ be the one having as basis all open intervals $(a,b)$ and all sets of the form ...
3
votes
4answers
293 views

Little confusion about connectedness

Consider $X=\{(x,\sin(1/x)):0<x<1\}$. Then clearly $X$ is connected , as it is a continuous image of the connected set $(0,1)$. So, $\overline X$ is also connected , as closure of connected set ...
0
votes
1answer
36 views

Fundamental groups in path-connected space

I'm studying Fundamental groups and today I saw the follow theorem: Theorem: Let be $X$ a topological space path-connected and $x,y\in X$. Then, the application $\psi:\pi_1(X,x)\to \pi_1(X,y)$ is a ...
0
votes
1answer
46 views

How many connected components does the punctured cone of isotropic vectors have?

Consider a real vector space $T$ of dimension $p+q$ with a non-degenerate symmetric bilinear form, $B:T\times T\to\mathbb{R}$, with signature $(p,q)$. Define the cone $$ ...
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0answers
5 views

Conservative field $F$ on not simple connected set

Give an example of a field $F:D\subseteq \mathbb R^2 \to \mathbb R^2$ such that $D$ is a doubly connected set (that is $D$ has on "hole") but $F$ is conservative And if $D$ is a triply connected ...
0
votes
1answer
48 views

Proof of the Harnack inequality

Let $\Omega\subseteq\mathbb{R}^n$ be a domain, $\Omega'\subset\subset\Omega$ be a domain and $u\in C^0(\overline{\Omega})$. Suppose we know $$\sup_{\Omega'}u\le 3^n\inf_{\Omega'}u\tag{1}$$ if ...