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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1answer
16 views

Connected sets definition

Defn: A set $X$ is connected if there do not exist non-empty, disjoint open sets $U,V$ s.t $U$ $\cup$ $V$ $=X$. I thought intuitively that this meant that this was like the English dictionary ...
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1answer
30 views

Let $S=[0,1) \cup [2,3]$ and $f:S \to \Bbb R$ be a strictly increasing map such that $f(S)$ is connected. Which of the following statements is true?

$f$ has exactly one discontinuity. $f$ has exactly two discontinuities. $f$ has infinitely many discontinuities. $f$ is continuous. I know theorems related to connectedness and ...
0
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1answer
37 views

If $(X,d_1)$ and $(X,d_2)$ two connected metric spaces if only if $X\times Y$ is connected metric space

$(X,d_1)$ and $(X,d_2)$ are two connected metric spaces if and only if $X\times Y$ is a connected metric space with metric $$ D((x_1,y_2), (x_2,y_2)) = \max(d_1(x_1,x_2),d_2(y_1,y_2)).$$ I know that ...
1
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1answer
19 views

Example where Alexandroff compactification $X^*$ is connected but the initial space $X$ is not

Let $(X,\tau)$ be a topological space that admits a one-point compatification $(X^{*},\tau)$ (Alexandroff compatification). I know that if the space $X$ is connected, then $X^*$ is connected as well. ...
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2answers
37 views

Map from circle to real line

I am asked to show that, for any continuous $\phi:\;S^1\to\mathbb{R}$ where $S^1=\{ \|\mathbf{x}\|=1,\;\mathbf{x}\in\mathbb{R}^2\}$, there exists $\mathbf{z}\neq 0$ such that: ...
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4answers
36 views

(i) $\{(x,y) \in \mathbb{R}^2 |\;xy = 1\}\,\bigcup\, \{(x,y) \in \mathbb{R}^2 |\;y = 0\}$ is not connected

I need to understand the following (i) $\{(x,y) \in R^2 |\;xy = 1\}\;U \{(x,y) \in R^2 |\;y = 0\}$ is not connected however (ii) $ Y = \{(x,y) \in R^2 |\;x^2 + y^2 < 1\}\;U \{(x,y) \in R^2 |\;y ...
1
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1answer
27 views

Is $\{(1,0),(0,0)\}\cup\bigcup_{n\neq1}\{(x,\frac{1}{n}):x\in\Bbb{R}\}$ locally connected?

Let $X=\{(1,0),(0,0)\}\cup\bigcup_{n\neq1}\{(x,\frac{1}{n}):x\in\Bbb{R}\}$. Determine whether or not $X$ is locally connected and find its components. Well, I know that a space $X$ is said ...
1
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1answer
24 views

Connected spaces minus proper subspaces is connected

So, I have a topology problem here. It goes like this. We have X, Y conected topological spaces and A, B proper subspaces of X and Y respectively. I have to show that $X \times Y - A \times B$ is ...
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2answers
53 views

If $\{E_\alpha\}$ is connected, $\bigcap\limits_{\alpha\in A}E \neq \emptyset$, then $\bigcup\limits_{\alpha\in A}E$ is connected

If $\{E_\alpha\}_{\alpha\in A}$ is connected in $\mathbb{R}^n$, $\bigcap\limits_{\alpha\in A}E_\alpha \neq \emptyset$, then $\bigcup\limits_{\alpha\in A}E_\alpha$ is connected. I have zero intuition ...
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2answers
36 views

If $A$ and $B$ are conneted and $A\cap B\neq \emptyset$, then $A\cup B$ is connected

Can you please let me know if my proof is reasonable? Prove: If $A$ and $B$ are conneted in $\mathbb{R}^n$ and $A\cap B\neq \emptyset$, then $A\cup B$ is connected Proof: Suppose that $A\cap B$ is ...
2
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1answer
77 views
+50

A connectivity-preserving function from a connected set onto an interval

Let $C$ be a connected set in the plane and $I$ the unit interval interval. Call a function $f$ from $C$ onto $I$ Connectivity-preserving if the following is true for every subset $I'\subseteq I$: ...
1
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0answers
12 views

Showing continuity of a real valued function [duplicate]

Let $S=[0,1)\cup [2,3]$ and let $f:S\to \mathbb{R}$ be strictly increasing such that $f(S)$ is an connected subset of $\mathbb R$. How to show that $f$ is continuous? $f(S)$ is connected means it is ...
0
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1answer
17 views

Why is semi-locally simply connected defined this way?

I would like to know why we define a space $X$ to be semi-locally simply connected if $\forall p\in X \exists U\ni p: i(\pi_1(U))=0\subset \pi_1(X)$ (SLSC), where $i$ is induced by ...
0
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1answer
14 views

path connected componenet and connected components in a Lie group coincide

does anyone have an explanation\proof as to why path connected components and connected components of Lie groups coincide?
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1answer
29 views

Understanding a proof about nested nonempty connected compact subsets

I know this question has asked to death here on MSE but I have not found a satisfactory solution. A solution found online is extremely elegant but I do not quite understand it! Given nested ...
0
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1answer
34 views

What does it mean for the empty set to be connected and totally disconnected?

I am trying to prove that the empty set is disconnected, but every single post I can find on this topic is about showing empty set is connected. Recall definition of connected. A set $S$ is connected ...
9
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1answer
177 views

Is It Always Possible to Cross a Surface Exactly Once?

Yesterday, in my physics class, the following question arose: Is there a closed surface embedded in $\mathbb R^3$ dividing space into two connected components such that all paths from one ...
1
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1answer
33 views

When is $\cap_{i=1}^{\infty} A_i$ non-empty

$A_1\supset A_2\supset\cdots A_n\supset A_{n+1}\supset\cdots$ be an infinite sequence of non-empty subsets of $\mathbb R^3$.Which one of the following ensures that their intersection ...
7
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4answers
411 views

A kind of converse to the Jordan curve theorem

The Jordan curve theorem in $\mathbb{R}^2$ says that if $S$ is a closed curve in $\mathbb{R}^2$. Then $S$ splits $\mathbb{R}^2$ into exactly two connected components $A$ and $B$. I was thinking about ...
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2answers
44 views

Can someone help me understand proof that “a sphere is connected”

The question is to prove Prove $S = \{(x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2 = 1\}$ is connected. The answer provided as: The sphere can be represented as a union of meridians (each of ...
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2answers
21 views

Connected/No connected set in $\mathbb{R}$, application of Munkres definition.

in $\mathbb{R}$, with the usual topology, is the set $A = \mathbb{[a,b]}$ connected? what about $B = [a,b] \cup [c,d]$ where $a < b < c < d$, i would say $A$ is connected while $B$ is not. ...
0
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1answer
35 views

$(0,1]$ is connected in relative topology. Different proof

The interval $X=(0,1]$ is connected in $\mathbb{R}$ w.r.t the relative topology. I am trying to show that $\emptyset, X$ are the only subsets which are both open and closed (I have seen the direct ...
0
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1answer
58 views

A 'weird' topology

I've got some questions about the following situation, and some proof-verification requests. Let $\mathcal{T}$ be the smalles topology on $\mathbb{R}$ with the property that ...
4
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2answers
82 views

If $X \setminus A$ is disconnected then prove or disprove $X \setminus B$ is also disconnected

Let $X$ be a connected metric space ( with more than one point ) and $A \subseteq X$ be not closed in $X$ and such that $X \setminus A$ is not connected ; then is it true that $X \setminus B$ is also ...
2
votes
1answer
79 views

Does every connected metric space $X$ contains a connected subset $A$ such that $X \setminus A$ is infinite?

Convention : Whenever we are going to talk about connected spaces , we will mean with more than one point . I am trying to see whether every connected metric space $X$ contains a connected subset ...
6
votes
5answers
87 views

Existence of a special kind of continuous injective function $f\colon A \to \mathbb R$, where $A$ is countable, relating to connectedness

Let $A \subseteq \mathbb R$ be a countable set ($A$ induced with usual subspace topology), then does there necessarily exist a continuous injective function $f\colon A \to \mathbb R$ such that for ...
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3answers
42 views

Is the boundary of every compact connected subset ( with more than one point ) of $\mathbb R^2$ is Homeomorphic with $S^1$ ?

Is it true that the boundary of every compact connected subset ( with more than one point ) of $\mathbb R^2$ is Homeomorphic with $S^1$ ? I was thinking that union of two closed balls touching ...
1
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2answers
33 views

Open and connected set in metric space [duplicate]

In a normed space, we know that if a set is open and connected, it is path connected. Is it true for general metric space or general topological space?
2
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1answer
57 views

Existence of a special type of injective function $f:A \to \mathbb R$ , where $A \subseteq \mathbb R$ is uncountable

Does there exist an uncountable subset $A \subseteq\mathbb R$ such that there exist an injective function $f:A \to \mathbb R$ such that for every $a \in A$ , there exist a connected subset $S ...
4
votes
3answers
58 views

If a continuous path $\Xi$ in $A\subset\mathbb{R}^2$ starts and end on $\partial(A)$, show that $A-\Xi$ is disconnected

If a continuous path $\Xi$ in a closed and bounded subset $A\subset\mathbb{R}^2$ starts and end on $\partial(A)$, show that $A-\Xi$ is disconnected To make things formal let $T=[0,1]$ and say that ...
0
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0answers
12 views

On existence of continuous mappings mapping open bounded path-connected sets onto open bounded path-connected sets

I was just thinking a bit and this question somehow arrived, I will ask it below after some preparation. Let us deal with this problem in $\mathbb R^n$. Now, suppose that we choose some two subsets ...
2
votes
2answers
29 views

Existence of a special type of injection $f:\mathbb N \to S^1$ such that for every $n\in \mathbb N$ relating to connectedness

Does there exist an injective function $f:\mathbb N \to S^1$ such that for every $n\in \mathbb N$ , there exist a connected subset (depending on $n$) $A \subseteq S^1$ with more than one point such ...
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0answers
36 views

Continuous function on connected set is constant

I have asked one question previously differentiable map on a connected open set which says : $f$ is differentiable on $E$ and $E$ is open, conencted and $f'(x)=0$ for every $x\in E$ then $f$ is ...
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0answers
62 views

Weak, Regular, and Strong connectivity in directed graphs

There are 3 types of connectivity when talking about a directed graph $G$. 1) weakly connected - replacing all of $G$'s directed edges with undirected edges produces a connected (undirected) graph. ...
1
vote
1answer
37 views

Show that $O(n)$, the set of orthogonal n x n matrices is not connected

I want to show that $O(n)$, the set of orthogonal $n \times n$ matrices is not connected. I know that a connected space $X$ does not split into disjoint non-empty open subsets, so to prove $O(n)$ ...
2
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2answers
59 views

Is a Normed Vector Space Necessary to Prove Path Connectedness?

Path connectedness seems to be defined in a topological space, but can the existence of a path be proven without using the functions of vector addition, scalar multiplication and norm ? For example, ...
1
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1answer
40 views

Prove $S^1$ is not homeomorphic to $S^2$ using connectedness

I have to prove that the unit circle $S^1$ is not homeomorphic to the sphere $S^2$ using connectedness. Intuitively I know this is true, but I'm not sure how to prove this.. Can someone help me?
2
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0answers
46 views

Separation axioms and when does there exist a connected space all whose proper connected subspaces are homeomorphic to the whole space?

This question is in the spirit of this question Does every non-singleton connected metric space $X$ contains a connected subset (with more than one point) which is not homeomorphic with $X$? ; ...
2
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0answers
25 views

How to Show Path Connectedness of Double Cone

I am asked to show that the set $C=\{(x,y,z)\in \mathbb{R}^3 : x^2+y^2=z^2\}$ is path connected. Visually this makes sense to me given that the cones share the point $(0,0,0)$ which provides me with ...
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1answer
49 views

Find a metric $d$ on $X$ such that $(X, τ^{(d/X)})$ is not connected

$X = (\{0\} \times [-1,1] \cup \{(x,\sin(π/x)) : x \in (0,1]\} \subset \mathbb{R}^2$ Find a metric $d$ on $X$ such that $(X, τ^{(d/X)})$ is not connected. Note: $τ^{(d/X)}$ denotes the metric ...
4
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3answers
54 views

Closed Subset of Connected Space with Boundary a Single Point is Connected

Old qual question here. Let $X$ be connected and $C\subset X$ be closed such that the boundary of $C$ is a point. Show that $C$ is connected. Attempt: For contradiction, suppose $U,V$ are ...
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1answer
40 views

Break the connection between two edges by removing the minimum amount of edges

We know that a drug dealer is going to deliver from city A to city B. As the police, we want to avoid the delivery. Cities are connected to other cities by roads (undirected edges). We can place one ...
2
votes
1answer
51 views

Set of the vertex sets to make connected graph into disjoint sets of vertices?

Suppose a non-directed graph G with vertices V and paths P. What is the name for the vertex sets to make break the graph by removal of some vertices?
0
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1answer
41 views

The union of a set with its complementary components that have compact closure [closed]

Let $X$ be a topological space and $A$ be a subset of $X$. Let $A'$ be a subset of $X$ such that $A'$ contains $A$ and all those subsets of $X\setminus A$ which are maximal connected sets and ...
0
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3answers
33 views

Range of continuous function related

For a continuous function $f:\mathbb{R} \rightarrow \mathbb{R}$,let $Z(f)=\{x\in \mathbb{R}:f(x)=0 \}$. Then which of the following is true for $Z(f)$ $Z(f)$ is always open. $Z(f)$ is always ...
0
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1answer
33 views

An open $\Omega\subseteq\Bbb R$ with $V=E\cap\Omega$ and $U\subset\Omega^c$, where $U,V$ are relatively open and separate $E\subseteq\Bbb R^n$

The following statement comes from page 253 of the book An Introduction to Analysis, 3rd. ed., by William R. Wade. *8.38 THEOREM. Let $E\subseteq R^{n}$. If there exist nonempty, relatively open ...
2
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1answer
63 views

Show that $\overline A\cap B=A\cap \overline B=\emptyset$

Definition Let $\left(X,d\right)$be a metric space,and let$Y$ be a subset of$X$.A subset $E$ of $Y$ is relatively open in $Y$ if and only if there is an open subset $U$ in $X$ with $E=U\cap Y.$ ...
2
votes
2answers
41 views

If $\hat A= A \cup \left\{\right.$ connected components of $X-A$ which are relatively compact in $X\left.\right\}$, then for every $A \subseteq X$

(Here, $B$ is relatively compact means the closure of $B$ is compact.) $\hat A$ is compact. $\hat A=\hat {\hat A}$. $\hat A$ is connected. $\hat A=X$. I try to eliminate the options ...
0
votes
1answer
53 views

$X,Y$ be connected ; $f:X\to Y$ be a continuous function which is right-cancellative w.r.t. continuous maps on connected spaces ; is $f$ surjective? [closed]

Let $X,Y$ be connected topological spaces and $f:X\to Y$ be a continuous function such that for any connected space $Z$ and any continuous functions $g_1,g_2:Y \to Z$ , $g_1 \circ f=g_2 \circ f ...
4
votes
3answers
435 views

Does a connected countable metric space exist?

I'm wondering if a connected countable metric space exists. My intuition is telling me no. For a space to be connected it must not be the union of 2 or more open disjoint sets. For a set ...