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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20 views

Connected components and showing subsets are equal

If $Z_1, Z_2 \subset X $ are connected components, show that $Z_1 = Z_2$ or $Z_1 \cap Z_2 = \emptyset$ Note: we defined connectedness as a splitting of two open sets $U_1$, $U_2$ such that$U_1 = X$ ...
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0answers
13 views

Find Connected Components

Let $X = \mathbb{R^2}$ and let $F_k$ be the closed line segment joining $(-1,2^{-k})$ and $(1,2^{-k})$ Additionally let $a = (-1,0) \ ,\ b=(1,0)$ Consider $A = \{a,b\} \cup \ ...
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1answer
20 views

Prove this is a Connected Component

Let $Y$ = $ \{(x,y) \in \mathbb{R^2}\ |\ xy = 0\ \ \cap\ (x,y) \neq (0,0)\ \}$ Consider $U_+ = \{\ (x,0) \in \mathbb{R^2}\ |\ x>0\ \}$ I'm trying to show that $U_+$ is a Component of $Y$ My ...
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0answers
30 views

Help with proving the following: Open set $U$ in normed space $X$ is connected iff it is connected by polygonal lines.

Help with proving the following: Open set $U$ in normed space $X$ is connected iff it is connected by polygonal lines: $\forall a,b \in U , \exists P_n \subseteq U , P_n(a,b) $ I would like if ...
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1answer
46 views

Prove that $\{(x,y)\mid x\in\mathbb Q\}\cup\{(x,y)\mid y\in\mathbb Q\}$ is a connected subset of $\mathbb R ^2.$ [duplicate]

In my notebook, something is very briefly, not in detail whatsoever, path connectedness mentioned, and two assumptions are made about $x_1, x_2 \in \mathbb Q.$ If anyone can prove this I would greatly ...
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4answers
73 views

Proving that a continuous $f:X \to Y ; \ X,Y- \text{topological spaces}$ and $A \subseteq X ; A \text{ connected} \implies f(A) \text{ connected}$

Proving that $f:X \to Y ; \ X,Y- \text{topological spaces}$ and $A \subseteq X ; A \text{ connected} \implies f(A)-\text{connected}$ The answer is given like this just one step I do not understand ...
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2answers
76 views

A ring is a connected set

I not know how to prove this: For example $$A=\{(x,y,z)\in \mathbb{R^3}\mid 1 < x^2 + y^2 + z^2<2 \}$$ I know that $$\partial A=S(0,1)\cup S(0, \sqrt{2})$$ can that help me at all? I was also ...
1
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1answer
67 views

Is simply connectedness is a topological property?

A topological space $X$ is called simply-connected if it is path-connected and any continuous map $f:S^{1} \to X$ (where $S^1$ denotes the unit circle in Euclidean 2-space) can be contracted to a ...
1
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1answer
54 views

Locally connected Locally compact separable metric space

Let $X$ be a locally connected locally compact separable metric space. Is it possible to find a countable collection $\mathcal{B}$ such that every member of $\mathcal{B}$ is a nonempty peano subspace ...
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1answer
47 views

With what additional assumption, would a connected space be path connected?

Let $X$ be a connected space. What additional condition on $X$ would imply that $X$ is path-connected? The only one I know is by assuming $X$ is an open subspace of a normed space $V$. What else ...
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1answer
57 views

$X$ is locally connected and $f: X \to Y$ is onto where $Y$ has the quotient topology. Prove that $Y$ is locally connected

Those questions about connectivity drive me crazy - I'm having so much difficulty proving them. Say $X$ is a locally connected space, and $f: X \to Y$ is onto where $Y$ has the quotient topology. ...
2
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2answers
39 views

Connected Lie group is second countable?

I know this is true from various sources, unfortunately none of them give the full proof. I already have a start: Let $G$ be connected Lie Group. Choose $K$ to be any compact neighbourhood of the ...
7
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1answer
92 views

$X$ is connected, $A \subset X$ connected, and $C$ a component of $X\backslash A$. Is $\overline A \cap \overline C \ne \emptyset$?

I'm trying to prove or disprove the following statement: If $X$ is connected, $A \subset X$ is connected, and $C$ a component of $X\backslash A$ then $\overline A \cap \overline C \ne \emptyset$. I ...
3
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1answer
43 views

Showing pre-image under entire function is simply connected.

I am currently working on the following problem and have run into a bit of trouble: Consider an entire function $f$ s.t. $\overline{B_1(0)}\subset f(\mathbb{C}).$ Show that V, a component of ...
3
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1answer
42 views

Prove that if the closure of each open ball in compact metric space is the closed ball with the same radius, then any ball in this space is connected

I'm having some difficulty with the following problem in general topology: Prove that if the closure of each open ball in compact metric space is the closed ball with the same center and radius, then ...
2
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0answers
34 views

If base+fibres are connected, is the total space connected?

I have a morphism $f: X \to Y$ of smooth varieties over an alg. closed perfect field $k$. If $Y$ is connected and all the fibres of $f$ are connected, does it follow that $X$ is also connected? ...
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0answers
84 views

An open and connected subset $U\subseteq \mathbb C$ is still connected if you remove a curve that lies entirely in $U$

Let $U\subseteq \mathbb C$ be open and connected. If $f:[0,1]\rightarrow U$ is continuous with $f(0)\neq f(1)$ and $f(s)\neq f(t)$ for $s\neq t$, then $U\setminus f([0,1])$ is connected. This seems ...
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1answer
33 views

Unit quaternion ball is compact and connected?

Let$$\mathbb{U} := \{x \in \mathbb{H} : |x| = 1\}.$$This is a group under multiplication. What is the easiest way to see that $\mathbb{U}$ is a compact and connected subset of $\mathbb{H}\cong ...
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1answer
31 views

Hausdorff space definition in terms of disconnected subsets

I've read the definition of a topological Hausdorff space (two distinct points have disjoints neighbourhoods) and of a disconnected space (it is the union of two disjoint nonempty open sets) Now I ...
3
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0answers
65 views

$R^2\setminus K, K$ compact, is not simply connected

So, I think I'm aware of the general idea of how to do this. $K$ is compact, hence closed and bounded, so there is some circle of finite radius, say $r$, that wraps around $K$. This loop isn't null ...
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0answers
25 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 ...
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1answer
22 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 $N$ ...
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1answer
19 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 ...
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. ...
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3answers
152 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
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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 ...
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2answers
24 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 ...
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0answers
43 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
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1answer
65 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
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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. ...
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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 ...
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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 ...
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2answers
80 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
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1answer
73 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
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1answer
26 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
44 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
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1answer
33 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
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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 ...
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2answers
89 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 ...
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4answers
93 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 ?
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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
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1answer
45 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
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1answer
58 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
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2answers
26 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 ...
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6answers
542 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 ...
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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. ...
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1answer
238 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
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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 ...
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2answers
71 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 ...