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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How many connected components can the intersection of a ball with an arbitrary domain have?

Let $\Omega \subseteq \mathbb{R}^n$ be an arbitrary domain (i.e. a connected and open subset). Let $x \in \Omega$ and $B_r(x)$ be the ball (w.r.t. euclidean metric) around $x$ with radius $r$. How ...
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0answers
43 views

Proving the following set is connected

I'm trying to solve this problem and it would be awesome if you could help me. Consider the sub-metric space of the normed space $(R^2,N_\infty)$, where $$ D = A\cup B\cup C $$ where $$ A = \bigcup_{...
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1answer
39 views

A Bending Buzz Wire Game

There is a wire connecting an exit and an entry point. At the entry, the wire has height $0$, at the exit, it has height $1$. Since the wire is connected, the wire has height $1/2$ somewhere, whatever ...
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2answers
50 views

Map from 2-sphere into $(\mathbb R^3, |\cdot|)$ [duplicate]

Can you help me with this? Let $S^2 := \{x\in \mathbb R^3:||x||_2 = 1\} \subset (\mathbb R^3, ||\cdot||_2)$ and $T:S^2 \to (\mathbb R, |\cdot|)$ be a continuous map. Since $S^2$ is compact, $T$ ...
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0answers
54 views

$X := \prod_{i\in I} X_i \: \: $ Show that X is (path-)connected, if $X_i$ is (path-)connected $\forall i \in I$ [on hold]

Let $I$ be an indexset and $(X_i, \mathcal T_i)$ a topological Space for $i\in I$. Let $X = \prod_{i\in I} X_i$ have the product Topology. Now i have to show the following two things: $X$ is ...
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0answers
22 views

Prove that the cantor space is totally disconnected.

Prove that the cantor space is totally disconnected. Let $(G,T)$ be the Cantor space and let $\prod_{i=1}^{\infty}(A_i,T_i)$ be homeomorphic to the Cantor space where $(A_i,T_i) = (\{0,2\}, T_{...
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2answers
30 views

Paths in a connected subset of $\mathbb R^n$

Prove that every connected subset $X$ of $\mathbb R^n$ with more than one point has the continuum cardinality. In order to use Baire Lemma, I would like to demonstrate that there is a compact (or ...
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2answers
33 views

Proof of 2-sphere being connected [duplicate]

Is there a short proof for the 2-sphere being connected? I only saw proof for the n-sphere but that is more complex than I need it to be.
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1answer
66 views

Complement of a simply connected set is simply connected

I saw the following surprising statement in Wikipedia: When $D\subseteq\Bbb C$ is a simply connected compact set, then its complement $E=D^c$ is a simply connected domain in the Riemann sphere ...
4
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1answer
49 views

Characterizing spaces with no nontrivial covers

I know that simply connected locally path-connected spaces have no nontrivial covers. Is there a characterization of spaces with this property?
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0answers
44 views

Connected vs. Necessarliy connected.

Let $A$ and $B$ be connected subsets of a topological space $(X,T)$ such that $A \cap B \neq \emptyset$. $(i)$ If $(X,T) = \Bbb R$, prove that $A \cap B$ is connected. $(ii)$ If $(X,T) = \Bbb R^2$, ...
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1answer
23 views

Space is connected but not locally connected

Let $(X,T)$ be the subspace of $\Bbb R^2$ consisting of the points in the line segments joining $<0,1>$ to $<0,0>$ and to all the points $<\frac{1}{n}, 0>, n = 1, 2, ...$. Show that $...
4
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1answer
59 views

To characterize uncountable sets on which there exists a metric which makes the space connected

For which uncountable sets $X$ is it true that there exist a metric $d$ on $X$ such that $(X,d)$ is connected ? [ The motivation for this question is : I wanted to characterize function $f : X \to X$...
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4answers
63 views

Show that the unit sphere is connected [duplicate]

I need to show that $\{(x,y,z)\in\mathbb{R}^{3}:x^2+y^2+z^2 = 1\}$ is connected. Intuitively I understand that it is path connected and, therefore, connected. However, I don't understand how I would ...
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3answers
56 views

$f:S^1 \to \mathbb R$ be continuous , is the set $\{(x,y) \in S^1 \times S^1 : x \ne y , f(x)=f(y)\}$ infinite ?

Let $f:S^1 \to \mathbb R$ be a continuous function , I know that $\exists y \in S^1 : f(y)=f(-y)$ where $y \ne -y $ (since $||y||=1$) , so that the set $A:=\{(x,y) \in S^1 \times S^1 : x \ne y , f(x)=...
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1answer
29 views

If union and intersection of two subsets are connected, the subsets are connected

I've been able to prove what is proved here If union and intersection of two subsets are connected, are the subsets connected? However, I was wondering if I could get some help finding an example to ...
2
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1answer
34 views

Showing this set is not path connected

Show that the region $\{(x,y) \in \mathbb{R}^2 : x< 0 \ \text{or} \ x>1 \}$ is not path connected. Suppose that $\{(x,y) \in \mathbb{R}^2 : x< 0 \ \text{or} \ x>1 \}$ is path connected. ...
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2answers
306 views

Can Path Connectedness be Defined without Using the Unit Interval?

Can path connectedness be defined without using the unit interval or more generally the real numbers? I.e., do we need Dedekind cuts or Cauchy convergence equivalence classes of the rational ...
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1answer
19 views

Image of a disconnected set is disconnected

I'm aware that the image of a connected set is connected and the preimage of a disconnected set is disconnected. However, I'm struggling to find an example of a disconnected set such that the image of ...
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1answer
34 views

General method: show subset of $\mathbb{C}$ is connected

Consider the two sets $$ A = \{z \in \mathbb{C} : |z^2 - 3| < 1\}, ~~~~ B = \{z \in \mathbb{C} : |z^2 - 1| < 3 \} $$ $B$ is connected, while $A$ is not. However, I have no idea how to prove this....
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1answer
53 views

Difficulty in choosing correct answer among the options.

1) The Cantor set, a subset of the real numbers: A. is not compact. B. is not contained in an interval. C. does not contain a non-trivial interval. D. does not have uncountably ...
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2answers
213 views

Is Every (Non-Trivial) Path Connected Space Uncountable?

I know that every non-trivial metric space with more than one point which is connected is uncountable. However, if we don't demand that the space be a metric space, we can find examples of such odd ...
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0answers
19 views

Equivalent definitions for simple connectivity

For a path connected space $X$, it is simply connected iff any two paths sharing endpoints are homotopically equivalent. Here simple connectivity means it has trivial fundamental group. I'm doing ...
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1answer
21 views

Locally connected iff every open cover has a connected refinement?

Is it true that a space is locally connected iff every open cover has a refinement into connected open sets? I know locally connected implies this, but I'm not sure about the converse. I'm having ...
2
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1answer
23 views

General approach to connectedness proofs, with specific example

Connectedness proofs have given me trouble for far too long. Does anyone have good insights on how to go about proving a set is connected? I know that is a broad question, any insight at all would ...
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1answer
25 views

Connectedness and path connectedness in $\mathbb R^2$

My task is to show $$ A = \{ (t, \sin(1/t)), t>0 \} \cup \{ (0,y), -1 \leq y \leq 1 \} \subset \mathbb R^2 $$ is connected and is not path connected. So far I realized that if it were path ...
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0answers
25 views

Nonexistence of a continuous injection from $\mathbb{R}^n$ to $\mathbb{R}$, for all $n \geq 2$. [duplicate]

I'm trying to do the following exercise from my lecture notes: There does not exist a continuous injection from $\mathbb{R}^n$ to $\mathbb{R}$, for all $n \geq 2$. I don't really know where to ...
3
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1answer
46 views

Is this proof of path-connected $\implies$ connected correct?

Let $U$, $V$ be two open disjoint sets over $X$ such that $X=U\cup V$. Let $p\in U$. For any $q\in V$, let $\alpha :[0,1]\rightarrow X$ be a path connecting $p$ and $q$. Then the set $A:=\alpha^{-1}(U)...
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1answer
39 views

Identifying some properties of a set

Let $S \subset R^2$ be defined by $S$ = {$(m+ \frac{1}{4^{|p|}} , n+ \frac{1}{4^{|q|}}): m,n,p,q \in Z$} Then, $S$ is discrete in $R^2$. The set of limit points of $S$ is the set {$(m,n) : m,n \in ...
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2answers
71 views

If $f: X \rightarrow Y$ is continuous, and $X$ is connected, then $f(X)$ is connected

I'm trying to understand the proof of: If $f: X \rightarrow Y$ is continuous, and $X$ is connected, then $f(X)$ is connected. What are we trying to do in the following proof (are we proving the ...
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1answer
34 views

Simply Connected sets

In my textbook it states, that the Union of two open docs is simply connected but not connected Why is this. I know simply connected means any closed path or loop can be shrunk to a point ...
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0answers
18 views

how to know whether a subset of euclidean space is path connected or not?(2)

I am asked whether $X=\{(x,y,z)|x^2+y^2−z^2=1\}\subset \mathbb{R^3}$,is path connected or not. i just know that $X$ is a closed subset.how can i answer this question? is there any hint? i asked a ...
0
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1answer
45 views

Is the unit sphere in $\Bbb R^4$ is path connected?

I am asked whether $X=\{(x,y,z,w)|x^2 + y^2 + z^2 + w^2 = 1 \}\subset \mathbb{R}^4$, is path connected or not. I just know that $X$ is a closed subset. How can I answer this question? Is there any ...
2
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1answer
35 views

If $A$ is a component in $(X,d_1) \iff$ $A$ is a component in $(X,d_2)$ then is $(X,d_1)$ homeomorphic to $(X,d_2)$?

Suppose that $A$ is a component in $(X,d_1) \iff$ $A$ is a component in $(X,d_2)$. Is it always the case that $(X,d_1)$ is homeomorphic to $(X,d_2)$? I have been trying to find a counter example, but ...
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2answers
56 views

Why is a circle not simply-connected?

To be simply-connected means to be path-connected and able to continuously shrink a closed curve while remaining in the domain. According to wikipedia, a circle is not simply connected, but a disk ...
2
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1answer
16 views

Find a connected graph that has exactly $2$ cutpoints of order $2$ and $3$ cutpoints of order $3$

Find a connected graph that has exactly $2$ cutpoints of order $2$ and $2$ cutpoints of order $3$ Definition: A cut point of order $k$ is a point $a \in X$ whose complement $X-\{a\}$ consists of $k$ ...
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2answers
50 views

Is this proof regarding product of connected spaces correct?

Let $X,Y$ be connected spaces, and consider their product $X\times Y$. I want to show that their product is connected. The posts I've read here regarding this question often include creating "slices" $...
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1answer
25 views

Why are the (connected) components of a topological space themselves connected?

I am trying to prove that (connected) components of a topological space are connected. I'll first define what I mean by a 'component of a topological space': For a topological space $X$, write $x\...
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1answer
30 views

Showing the Following Set is Path Connected

Would be grateful if someone could help me with the following question "For any normed space X which is NOT the set of Reals, the set X (excluding 0) is path connected In the solutions the usual ...
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1answer
27 views

Complex variable, multiplication of numbers

Question: Let a and b be complex numbers with $a \neq 0.$ Describe the set of points $az + b $ as $z$ varies over the first quadrant, $\{z = x+iy: x>0 \,and \,y>0\}$ Solution: Let $a = |a|e^{i\...
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3answers
62 views

Does there exist a continuous function $g:S^1 \to S^1$ such that $(g(z))^2=z , \forall z \in S^1$?

Let $S^1:=\{z \in \mathbb C:|z|=1\}$ ; does there exist a continuous function $g:S^1 \to S^1$ such that $(g(z))^2=z , \forall z \in S^1$ ?
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2answers
34 views

Closed unit square is connected.

Question. Prove that the closed unit square $S=\{x$ in $\Bbb R^2 : 0 \le x_1 \le 1, 0 \le x_2 \le1 \}$ is connected. I understood how to prove unit interval is connected and I am trying to extend to ...
4
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1answer
55 views

Connected components functor for free coproduct cocompletions

Any extensive category admits a notion of connected object and hence a disconnected object. However, not all disconnected objects are presentable as disjoint unions of connected objects. Among ...
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1answer
35 views

For every pair $n,d$ such that $d \ge (n-1)/2$ prove that $G$ on $n$ vertices with minimum degree $d$ is edge d-connected.

For every pair $n,d$ such that $d \ge (n-1)/2$ prove that $G$ on $n$ vertices with minimum degree $d$ is edge d-connected. None of my observations I was able to obtain seem to be useful. I am just ...
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1answer
47 views

Vertex connectivity of $K_n$ upon removal of edges of subgraph $C_n$

Consider graph $G$ = $K_n$ - $E(C_n)$ ($G$ is complete graph on $n$ vertices upon removal of edges of subgraph $C_n$). For every $n \ge 3$ find maximum $k$ such that $G$ is vertex $k$-connected. I ...
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1answer
30 views

$X \subseteq M(n,\mathbb C) ; |X|>1 ; $ connected/path connected, what about $S:=\{x \in \mathbb C : x $ is an eigenvalue of some matrix in $X\}$?

Let $X \subseteq M(n,\mathbb C)$ be a set with more than one element and $S:=\{x \in \mathbb C : x $ is an eigenvalue of some matrix in $X\}$. I know that if $X$ is compact then so is $S$. My question ...
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1answer
43 views

“Continuous maps are those maps that do not tear space apart”

In a tutorial I wanted to give a quick explanation of the property of continuity. One of the common intuitions for continuity is that it preserves connection: Continuous maps do not map connected ...
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1answer
43 views

Connectedness of the sets $\{A \in M(n,\mathbb R) : \mathrm{rank}(A) =m\}$ and $\{A \in M(n,\mathbb R) : \mathrm{rank}(A)\ge r\}$

Let $r>0$ , I know that $\{A \in M(n,\mathbb R) : \mathrm{rank}(A) < r\}$ is path connected in $M(n,\mathbb R)$ . My question is ; for positive integer $m < n$ , is the set $\{A \in M(n,\...
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vote
1answer
40 views

How does $\frac{1}{2}(n-s+1)(n-s)$ equal $\binom{n-s+1}{2}$?

Maybe a basic question, but I'm strolling through graph theory at the moment after a few years out of tertiary mathematics. There is a theorem that if a graph $G$ has $s$ connected components, then $$ ...
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0answers
54 views

On a function $f: \mathbb R^m \to \mathbb R^n$ , $n>1$ , mapping connected sets to connedted sets and discontinuous at a point

Let $f: \mathbb R^m \to \mathbb R^n$ be a function mapping connected sets to connected sets where $n>1$ ; let $a \in \mathbb R^m $ and $ \epsilon >0$ be such that $f(B_{\delta}(a)) \cap (\...