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Questions tagged [connectedness]

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.

2
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1answer
90 views

What's wrong with the following proof that any interval $(a,b) \subset \mathbb{R}$ is connected?

Suppose not. Let $U, V$ be nonempty disjoint open sets such that $(a,b) = U \cup V$. Let $x \in U, y \in V$, and w.l.o.g. assume that $x < y$. Consider the set $S = \{z \in V | z > x\} \subseteq ...
0
votes
1answer
15 views

Let A be a proper subset of X and B a proper subset of Y. If X and Y are connected, show that (XxY) $\setminus$ (AxB) is connected.

I know that this question is already answered in the site but I did it in a different way but I don't know if it is correct. My attempt: Suppose that $(X\times Y) \setminus (A\times B)$ is ...
1
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0answers
56 views

How to show that a set is connected

Can someone please explain to me the idea of connectedness and how they relate to these sets? I understand the definition of connectedness, but I am struggling to relate the definition to these ...
1
vote
1answer
35 views

Connected subset of subset is connected in whole space.

Soppose $(X,T)$ is a topological space. Let $Y\subset S\subset X$. Is $Y$ connected in $S$ iff it is connected in $X$. Possible Proof: Since $Y$ is connected in S the only closed and open sets in $...
3
votes
1answer
46 views

Is the set of bounded functions connected?

One of my topology homework questions this week says the following: Consider the space of bounded functions $B[0, 1]$ on the interval $[0, 1]$ with the $d\infty$ metric. Prove that in the ...
4
votes
0answers
44 views

Number of unique paths on the edges of a grid with wraparound that return to the origin

I was given this problem on the codegolf stackexchange, but I don't know where to begin on how to calculate it, except by creating some brute-force program to do it for me (like almost all existing ...
2
votes
1answer
57 views

Showing every connected regular space having more than one point is uncountable without using proof by contradiction

The common proof goes like this: Suppose $X$ is countable, then it must be Lindelöf. A regular Lindelöf space is normal (akin to the proof that a regular and second-countable space is normal). ...
0
votes
2answers
27 views

$(A_i)_{i\in E}$ family of connected sets such that $\bigcap\limits_{i\in E} A_i \neq \emptyset$ then $\bigcup\limits_{i \in E} A_i$ is connected

My idea so far is, since $\bigcap\limits_{i\in E} A_i \neq \emptyset$, then exists $p \in\bigcap\limits_{i\in E} A_i$ if $\bigcup\limits_{i \in E} A_i$ is not connected, then exists $A$ and $B$ open ...
1
vote
1answer
15 views

Completely disconnecting a separable metric space by removing a sequence of countable dense subsets

I am wondering if given a separable metric space $X$, it is possible to totally disconnect $X$ by repeatedly removing countable dense subsets. For example, let $I_1$ be a countable dense subset of $X$...
2
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1answer
45 views

Is connected component open?

There is a theorem that:A space is locally connected iff each connected components of an open set is open. But recently I had seen to prove That each connected component is closed. Connected ...
0
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2answers
40 views

Number of connected components Invariant

Why is the number of connected components invariant under homeomorphisms? I know that connectedness, as well as path connectedness, are properties conserved through homeomorphisms. But why is this ...
0
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0answers
36 views

A problem on path connectedness of unit ball in R*R

Actually, I am having a problem over the fact that as $f:[0,1]\rightarrow\mathbb{R}^{2}$ , $f(t)= (1-t)x + ty$ is continuous, $x,y\in\mathbb{R}^{2}$, then $f([0,1])$ is connected . Then as unit ball ...
-1
votes
2answers
49 views

How to show that $\infty$ is isolated?

A point $p$ in a topological space $Y$ is isolated if there exists an open set $O$ such that $p \in O$, but $(Y \setminus \{p\}) \cap O=\emptyset$ Suppose that $X$ is compact. Show that $\infty$ is ...
1
vote
1answer
31 views

Suppose that the sets $A_{1},A_{2} \subset \mathbb{R}^n $ are connected and that they are not disjoint. Prove that $A_{1} \cup A_{2}$ is connected.

Suppose that the sets $A_{1},A_{2} \subset \mathbb{R}^n$ are connected and that they are not disjoint. Prove that $A_{1} \cup A_{2}$ is connected. The section including this question contains this ...
0
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0answers
25 views

Generated Subgroup connected in topological group?

Let $G$ be a non-abelian, uncountable topological Hausdorff group (not necessarily connected) and $x \in G$. Then of course $\overline{\langle x \rangle}$ is a closed subgroup of $G$. My question is: ...
0
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1answer
20 views

G is matrix lie group such that there is $A\in G$, it is not written as $A=e^{X_1}…e^{X_m} $ for some $X_1,..X_m\in \mathfrak g$

Counterexample: G is matrix lie group such that there is $A\in G$ such that A can not be written as $A=e^{X_1}....e^{X_m} $ for some $X_1,..X_m\in \mathfrak g$ where $\mathfrak g $ is lie algebra of G....
0
votes
1answer
40 views

Minimal dimension of an affine space in $\mathbb R^n$ that could divide an open set $U$ into disconnected components?

Suppose $U \subseteq \mathbb R^n$ ($n \ge 3$) is an open, contractible set. I am thinking about what would be a minimal requirement on dimension for an affine subspace $\mathcal A$ to divide $U$ into ...
1
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1answer
28 views

understanding orientable manifolds

I'm reading Warner. "Foundations of Differentiable Manifolds and Lie Groups." p. 138. I don't get the statement in the definition of orientable manifolds. 4.1 Definitions $\;$ (the preface omitted) ...
1
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1answer
38 views

Proof of theorem of connectedness of an open set

I was watching an online lecture about complex analysis and in one if the first videos: The following theorem is stated: Let $G$ be an open set in $\mathbb{C}$. Then $G$ is connected if and only if ...
1
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1answer
36 views

The image of a continuous mapping on a connected metric space is connected: ($\epsilon - \delta$) proof

I've seen entirely set-theoretical solutions to this proof, but not many based on epsilon-delta arguments. I tried writing one, but am not sure whether my arguments hold. Theorem: The image of a ...
1
vote
1answer
44 views

What does this equation imply?

I am reading a paper on seam carving to implement, and got stuck in understanding the convention used in the equation below. 1) I understand, $\{ ( x ( i ) , i ) \} _ { i = 1 } ^ { n } , \text { s.t....
0
votes
1answer
47 views

Proving the Daisy Lemma [duplicate]

Lemma: Suppose that $A,B \subseteq X$ are connected and $A \cap B \neq \emptyset$ , then $A \cup B$ is connected. How would I go about proving this? I think I understand the consequences of the lemma ...
1
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3answers
41 views

Compact or open $\{0\}\cup\{\frac1n + \frac1m | m,n \in N\})$ in R?

Compact or open $\{0\}\cup\{\frac1n + \frac1m | m,n \in N\})$ in R ? The question is straight forward There exists no interval about $2\in S$ that has only elements of S. Not open What about ...
0
votes
0answers
55 views

de Rham cohomology and homotopy groups of differential manifold

As is well known, a differential manifold has trivial degree-$0$ de Rham cohomology $H^0(M) = 0$ if and only if it is connected. It seems that the degree-$1$ de Rham cohomology group $H^1(M)$ being ...
3
votes
1answer
27 views

Is the subset of $\mathbb C^n$ that is conjugate invarivant and permutation invariant simply connected in the quotient space?

Suppose $\sim$ is a relation on $\mathbb C^n$ such that $x \sim y$ if and only if there is some permutation $\sigma \in S_n$ with $$ x = (x_1, \dots, x_n) = (y_{\sigma(1)}, \dots, y_{\sigma(n)}) = \...
3
votes
1answer
65 views

Extending $S^1$ to an embedding of $D^2$ in $\mathbb{R}^3$

Consider the specific embedding $f : S^1 \to \mathbb{R}^3$ given by, say, the unit circle in the $xy$-plane. Suppose further that this embedding is contained within a $3$-dimensional, simply-connected ...
2
votes
1answer
23 views

Showing that $g(T)\subseteq f'(I) \subseteq\overline{g(T)}$ where $g(x,y) = \frac{f(x) - f(y)}{x-y}$

I'm trying to solve a three part exercise. The first part asks me to show that the set $T=\{(x,y)\in I\times I : x < y\}$, where $I$ is just some open interval in $\mathbb{R}$ is connected. I've ...
2
votes
2answers
45 views

Prove that if a graph $G$ has a Hamilton path, then for every $S\subseteq V(G)$, the number of components of $G-S$ is at most $|S|+1$.

Prove that if a graph $G$ has a Hamilton path, then for every $S\subseteq V(G)$, the number of components of $G-S$ is at most $|S|+1$. My solution (rough and incorrect): Consider a Hamilton path $P$ ...
2
votes
1answer
26 views

Connectedness $A= \left\{ (x,mx) \mid m \in \mathbb{N}, x \in [ 0 , 1 ] \right\} \cup \left\{ (x,-mx) \mid m \in \mathbb{N}, x \in (0 , 1) \right\}$

Is set $$ A = \left\{ (x,mx) \mid m \in \mathbb{N}, x \in [ 0 , 1 ] \right\} \cup \left\{ (x,-mx) \mid m \in \mathbb{N}, x \in (0 , 1) \right\} $$ connected? I tried by dividing the set into two ...
0
votes
1answer
30 views

Problem : connectedness of sets

Are sets $$ A = \left\{ (x,y) \in \mathbb{R}^2 \mid -1 \leq y \leq \frac{2}{x^2 + 1} -1 \right\} $$ and $$B = \left\{ (x,y) \in \mathbb{R}^2 \mid y \geq x^2 -1 \right\} $$ connected? I think they are ...
-3
votes
2answers
40 views

A simple question about Theorem 2.47 on p.42 in “Principles of Mathematical Analysis 3rd Edition” by Walter Rudin.

I am reading "Principles of Mathematical Analysis 3rd Edition" by Walter Rudin. There is the following theorem on p.42: Theorem 2.47 A subset $E$ of the real line $\mathbb{R}^1$ is connected ...
6
votes
3answers
902 views

Is a connected set always an uncountably infinite set?

I'm trying to understand the concept of a connected set. The classic example which is presented is that of $\Bbb R$ or any interval of $\Bbb R$ with the usual topology. Moreover, I heard that to a ...
0
votes
1answer
29 views

Properties of this topology on $\Bbb X$.

For given the usual topology $\tau$ on $\Bbb{R}$, define the compact complement topology on $\mathbb{R}$ to be $$\tau'=\{A\subseteq \Bbb{R}:A^C\text{ is compact in }(\Bbb{R},\tau)\} \cup \{\emptyset \...
0
votes
3answers
44 views

Properties of this topology on $\mathbb N$.

$\mathbb N$ is consist of the basis generated by the set $A_n=\{n,n+1,n+2,n+3....\}$ . Then what properties does it have? Hausdorff: for any natural numbers $x$ and $y$ there are no disjoint open ...
2
votes
1answer
38 views

Is every finite, nondiscrete $T_0$ space connected?

Is every finite, nondiscrete $T_0$ space connected? What I've tried is to find a separation and thus get a contradiction with the statement that the space is nondiscrete. After some time I've got lost ...
0
votes
1answer
12 views

Help with Proof for Connect Open Space implying Finitely path connectedness

I would appreciate help with my (sketch of a) proof for the following problem: Question: Let $E\subset \mathbb{R^n}$ be an open connected open set. Show that for any $x,y\in E$ there is a finite ...
1
vote
4answers
74 views

Intuition behind definition of Connected Space in Topology

There are 2 definitions of Connected Space in my lecture notes, I understand the first one but not the second. The first one is: $$\mbox{A topological space } (X,\mathcal{T}) \mbox{ is connected if ...
1
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1answer
49 views

Show that a set is connected

Show that $S:= \{ x+iy \;|\; x = 0 \text{ or } x>0 , y = \sin(\frac{1}{x}) \} $ is connected, even though there are points in $S$ that cannot be connected by any curve in $S$ Attempt: Suppose $S$ ...
6
votes
1answer
43 views

Examples of connected door spaces.

A topological space $X$ is a door space if any subset of $X$ is either open or closed (or both). Naturally, a connected door space is that in which any proper subset is either open or closed, but not ...
1
vote
1answer
57 views

Compactness and connectedness in $\Bbb R^3$

Consider the set $$A=\left\{ \begin{pmatrix} x\\y\\z\end{pmatrix} \in \Bbb R^3: z=x^2+y^2+1\right\} \subset \Bbb R^3$$ Prove of disprove: $A$ is connected and compact The set $A$ is unbounded, ...
1
vote
3answers
51 views

How to prove the set of all symmetric matrices with eigenvalues in $(0,2)$ is connected?

Prove that space $X$ of all symmetric matrices in $GL_2(\mathbb R)$ with both the eigenvalues belonging to the interval $(0,2),$ with the topology inherited from $M_2(\mathbb R) $ is connected. ...
6
votes
1answer
94 views

Compact spaces in which any closed set can be partitioned into finitely many closed sets whose clopen subsets extend to the whole space

Let $X $ be a compact topological space (not necessarilly Hausdorff). I am looking for a charactrization for the following property: Property: If $C $ is a closed subset of $X $, then there are ...
1
vote
1answer
54 views

Connected component of $\{(x_1,…,x_n) \; | \; x_1^2+…+{x_{n-1}}^2-x_n^2 \neq 0\}$

The question is in the title, actually for $n=2$, it's okay, for $n=3$, I thought that : $\{(x_1,...,x_n) \; | \; x_1^2+x_2^2-{x_{3}}^2 < 0 \}, \{(x_1,...,x_n) \; | \; x_1^2+x_2^2-{x_{3}}^2 > 0 ...
2
votes
2answers
50 views

Connected set in $C[0,1]$

Let $X=C([0,1])$ be the space of all continuous real valued function on $[0,1]$. Equip $X$ with the $\sup$ metric, $d(f,g)=\sup_{x\in[0,1]}|f(t)-g(t)|$ Let, $S=\{f\in X| f(0)\neq f(1)\}$ $T=\{f\in ...
2
votes
1answer
33 views

Proof that direct product of connected spaces is connected

I am trying to understand a piece of the proof that the direct product of connected spaces is itself a connected space, as given by Lee in "Introduction to Topological Manifolds". By induction, we may ...
1
vote
1answer
32 views

Where is my reasoning wrong?

We know that components of locally connected sets are open. Now consider the space $$X=\bigcup\limits_{q \in \mathbb{Q}}^{} \{(x,y) \in \mathbb{R}^2; x^2+y^2=q^2 \}-\{(0,0) \}$$ Now this seems locally ...
1
vote
0answers
24 views

connected subgraphs having minimum expansion

If I have a connected graph $G$, is it true that subgraphs(corresponding to cuts) having minimum conductance or expansion(or other connectivity measures) should be connected? In other words if $S$ is ...
1
vote
1answer
46 views

A path-connected graph is connected as a graph

This question regards the difference between being path-connected as a topological space, i.e. "Every two point $x,y \in X$ can be connected with a path" and being connected in a graph sense: "A graph ...
1
vote
1answer
46 views

IVT and connectedness on $\mathbb{R}^{n}$

Let $f: \mathbb{R}^{n} \to \mathbb{R}$ be a continuous function and $n \geq 2$. Moreover, suppose that there is $c \in \mathbb{R}$ such that $f^{-1}(\{c\}) = \{x \in \mathbb{R}^{n} \mid f(x) = c\}$ is ...
2
votes
2answers
98 views

Assume $A \subset \mathbb{R}^n$ is connected such that $A^c$ is separated by $B,C$, then $A \cup B$ is connected.

The following problem showed up on a previous qualifying exam: Assume $A \subset \mathbb{R}^n$ is connected such that \begin{equation}A^c = B \cup C, \text{ such that } \overline{B} \cap C = \...