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.

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333
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10k views

Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
153
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1answer
5k views

Is there a homology theory that counts connected components of a space?

It is well-known that the generators of the zeroth singular homology group $H_0(X)$ of a space $X$ correspond to the path components of $X$. I have recently learned that for Čech homology the ...
104
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2answers
4k views

Connected metric spaces with disjoint open balls

Let $X$ be the $S^1$ or a connected subset thereof, endowed with the standard metric. Then every open set $U\subseteq X$ is a disjoint union of open arcs, hence a disjoint union of open balls. Are ...
48
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2answers
864 views

'flimsy' spaces: removing any $n$ points results in disconnectedness

Consider the following property: $\mathbb R$ is a connected space, but $\mathbb R\setminus \{p\}$ is disconnected for every $p\in \mathbb R$. $S^1$ is a connected space and if we remove any point, ...
44
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4answers
2k views

Can a nowhere continuous function have a connected graph?

After noticing that function $f: \mathbb R\rightarrow \mathbb R $ $$ f(x) = \left\{\begin{array}{l} \sin\frac{1}{x} & \text{for }x\neq 0 \\ 0 &\text{for }x=0 \end{array}\right. $$ has a graph ...
44
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2answers
6k views

Path-connected and locally connected space that is not locally path-connected

I'm trying to classify the various topological concepts about connectedness. According to 3 assertions ((Locally) path-connectedness implies (locally) connectedness. Connectedness together with ...
40
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6answers
1k views

Is the complement of countably many disjoint closed 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 “disk&...
32
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4answers
4k views

Prove that the set of $n$-by-$n$ real matrices with positive determinant is connected

Math people: In the fourth edition of Strang's "Linear Algebra and its Applications", page 230, he poses the following problem (I have changed his wording): show that if $A \in \mathbf{R}^{n \times n}...
30
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3answers
1k views

Connected, locally connected, path-connected but not locally path-connected subspace of the plane

I am looking for a set in the plane (with respect to the natural Euclidean topology) that is connected, locally connected, path-connected but not locally path-connected. I did not find one in Steen-...
27
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7answers
11k views

Connected metric spaces with at least 2 points are uncountable.

That's a problem I proved (quite a while back) in tiny Rudin. However, I don't really get it. The other questions were actually useful results - I don't think I've ever come near using this result. ...
27
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1answer
362 views

Can a countable dense subset be split into two disjoint dense subsets?

Suppose $X$ is a compact connected Hausdorff space and $D \subset X$ countable and dense. Can we always write $D=D_1 \cup D_2$ as a disjoint union of countable dense subsets? More generally if $U \...
26
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3answers
549 views

Is the complement of an injective continuous map $\mathbb{R}\to \mathbb{R}^2$ with closed image necessarily disconnected?

I am interested in the following Jordan curve theorem-esque question: Suppose that you are given a continuous, injective map $\gamma: \mathbb{R}\to \mathbb{R}^2$ such that the image is a closed ...
25
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2answers
928 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 ...
25
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2answers
829 views

Is there a way to phrase “continuous” in terms of “connected”?

This is something of a pedagogical question. Most of us have scratched our heads at some point as to why the definition of "$f$ is continuous" requires that the preimage of any open set under $f$ is ...
25
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2answers
3k views

Connectedness of the boundary

My question is about the following claim: For $n \geq 2$, let $A\subset \mathbb R^n$ be a non-empty, open, bounded set. Assume $A$ and its complement are connected and $\text{int}(\text{cl}(A)) = A$...
24
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3answers
438 views

Connected topological space such that the removal of any of its points disconnects it into exactly $3$ connected components?

$\mathbb R$ has the property of being a connected space which is divided into $2$ connected components by the removal of any of its points. I'm trying to generalize this property by constructing a ...
24
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2answers
9k views

Topologist's sine curve is not path-connected

Is there a (preferably elementary) proof that the graph of the function $y$ defined on $[0,1)$ by $$ y(x) =\begin{cases} \sin\left(\dfrac{1}{x}\right) & \mbox{if $0\lt x \lt 1$,}\\\ 0 & \mbox{...
24
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2answers
2k views

Is bijection mapping connected sets to connected homeomorphism?

If $f : \mathbb{R}^{n} \to \mathbb{R}^{n}$ is a bijection, mapping connected sets to connected, is $f$ necessarily a homeomorphism? The converse is true, a well known property of homeomorphisms. I ...
23
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2answers
8k views

Topology of matrices

1.Consider the set of all $n×n$ matrices with real entries as the space $\mathbb R^{n^2}$ . Which of the following sets are compact? (a) The set of all orthogonal matrices. (b) The set of all ...
22
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6answers
4k views

How to show path-connectedness of $GL(n,\mathbb{C})$

Well, I am not getting any hint how to show $GL_n(\mathbb{C})$ is path connected. So far I have thought that let $A$ be any invertible complex matrix and $I$ be the idenity matrix, I was trying to ...
22
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5answers
20k views

Union of connected subsets is connected if intersection is nonempty

Let $\mathscr{F}$ be a collection of connected subsets of a metric space $M$ such that $\bigcap\mathscr{F}\ne\emptyset$. Prove that $\bigcup\mathscr{F}$ is connected. If $\bigcup\mathscr{F}$ is not ...
22
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1answer
530 views

Is every connected subset of the Sierpiński triangle arcwise connected?

I think this should be true. If it's indeed the case, it seems like this should be a known result, so references are welcome. I managed to prove that (assuming $S$ is the connected subset) $S$ minus ...
21
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2answers
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Why is the “topologist's sine curve” not locally connected?

In Munkres's Topology, it is claimed that "The topologist's sine curve" is not locally connected without further explanation (See Example 3 of Section 25 "Components and Local Connectedness", 2nd ...
20
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5answers
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Topologist's sine curve is connected

I just came across the example of the topologist's sine curve that is connected but not path-connected. The rigorous proof of the non-path-connectedness can be found here. But how can I prove that ...
20
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4answers
7k views

Showing that $\mathbb{R}$ is connected [duplicate]

So I know that $\mathbb{R}$ is both open and closed. But given a set, $X\subset \mathbb{R}$, $X\ne \emptyset $ that is both open and closed, how does one show that $X=\mathbb{R}$? Here is my ...
19
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9answers
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If a nonempty set of real numbers is open and closed, is it $\mathbb{R}$? Why/Why not?

In other words, are $\emptyset$ and $\mathbb{R}$ the only open and closed sets in $\mathbb{R}$? Why/Why not? I tried by assuming a set is equal to its interior points and contains its limit points. ...
18
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3answers
12k views

The closure of a connected set in a topological space is connected

This problem is from Rudin. I am trying to Prove that the closure of a connected set is always connected. Here is my proof. Let $E$ be a connected set in a space $X$. Suppose to the contrary that the ...
18
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1answer
749 views

Does path-connected imply simple path-connected?

Let $X$ be a path-connected topological space, i.e., for any two points $a,b\in X$ there is a continuous map $\gamma\colon[0,1]\to X$ such that $\gamma(0)=a$ and $\gamma(1)=b$. Note that beyond ...
18
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2answers
474 views

Finding simply connected open sets between compact ones and general open ones in $\mathbb R^2$.

In a paper I am reading (not a published one), the following is considered obvious: Let $K$ be a compact and connected subset of $\,\mathbb R^2$, with $\mathbb R^2\smallsetminus K$ connected, and $U\...
17
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1answer
132 views

Tracing a curve along itself - can the result have holes?

Let $\varphi:[0,1]\to\Bbb R^2$ be a continuous curve (not necessarily injective) with $\varphi(0)=(0,0)$. Let $f:[0,1]^2\to\Bbb R^2$ be defined as $f(s,t)=\varphi(s)-\varphi(t)$. Question: Is the ...
17
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1answer
383 views

Is the set $\{X \in \mathcal{M}({m \times n}) : \rho(M-NX) < 1\} $ connected?

Suppose $M \in \mathcal M(n \times n; \mathbb R)$ and $N \in \mathcal M(n \times m; \mathbb R)$ are fixed with $N\neq 0$. Let \begin{align*} E = \{X \in \mathcal{M}(m \times n; \mathbb R) : \rho(M-...
16
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2answers
808 views

Is there a canonical way to connect a topological space?

It is known that each topological space $X$ admits a Hausdorffization - which means that every topological space can be "approximated" by the corresponding unique Hausdorff space for which each ...
16
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5answers
11k views

Is there a short proof for the Intermediate Value Theorem

My final for my introductory analysis course is tomorrow and my teacher gave us a list of possible theorems to prove. If anyone could please show me a proof for The Intermediate Value Theorem that is ...
16
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2answers
4k views

connected manifolds are path connected

prove every connected manifold is path connected manifold . my thought: connected space : Let $ X$ be a topological space. A separation of $ X $ is a pair $U, V$ of disjoint nonempty ...
16
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0answers
503 views

Exercise in Engelking's book regarding a disconnected space.

This is related to 6.3.24 in Engelking's Topology book. The Hilbert space $H$ is the set of sequences $(x_i) \in \mathbb R ^\omega$ such that $\|x\|=\sum _{i=1} ^\infty x_i ^2<\infty$, with ...
15
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3answers
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What are some measures of connectedness in graphs?

I am not a mathematician (I am an engineer who is working on improving his mathematics), so I apologize in advance if my question is trivial. Consider a graph of $N$ nodes, with some defined ...
15
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3answers
657 views

If $C$ is a component of $Y$ and a component of $Z$, is it a component of $Y\cup Z$?

Let $X$ be a topological space, $Y$ and $Z$ subspaces of $X$. Let $C$ be a connected subset of $Y\cap Z$ such that $C$ is a component of $Y$ and a component of $Z$. Does it follow that $C$ is a ...
15
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1answer
532 views

Does the Jordan curve theorem apply to non-closed curves?

A Jordan curve is a continuous closed curve in $\Bbb R^2$ which is simple, i.e. has no self-intersections. The Jordan curve theorem states that the complement of any Jordan curve has two connected ...
14
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5answers
12k views

Product of connected spaces

You have two connected topological spaces $(A,B)$. Prove that $A\times B$ is also connected. I understand that I have to prove that there is a point in $B$ (call it $b$), that makes $A\times\{b\}$ ...
14
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5answers
7k views

Connected But Not Path-Connected?

Can you think of any spaces that are connected but not path connected apart from the Topologist's Sine Curve?
14
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2answers
2k views

Arcwise connected part of $\mathbb R^2$

Here's a question that I share: Show that if $D$ is a countable subset of $\mathbb R^2$ (provided with its usual topology) then $X=\mathbb R^2 \backslash D $ is arcwise connected.
14
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1answer
819 views

Is there a topological group that is connected but not path-connected?

Is there a $\big($T$_0$$\hspace{-0.02 in}\big)$ topological group that is connected but not path-connected? If yes: $\quad$ Can it be complete? $\:$ (with respect to the two-sided uniform structure) ...
14
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3answers
454 views

Metal Ball Cage Template Cardinality: A Brilliantly Lazy PROOF

N.B. - I'm looking for the simplest way to ascertain the number of templates $T$ (see below) comprising the structure from just one angle alone; that is, I'm sitting down looking up at this thing, and ...
13
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2answers
2k views

In a metric space X, if A is connected, is its interior connected?

I believe that int(A) is also connected. I tried to use an argument by contradiction, that is to say I supposed that int(A) is not conneted, but without success
13
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5answers
300 views

Showing that $\{x\in\mathbb R^n: \|x\|=\pi\}\cup\{0\}$ is not connected

I do have problems with connected sets so I got the following exercise: $X:=\{x\in\mathbb{R}^n: \|x\|=\pi\}\cup\{0\}\subset\mathbb{R}^n$. Why is $X$ not connected? My attempt: I have to find ...
13
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1answer
1k views

Complement of a totally disconnected compact subset of the plane

Let $E \subset \mathbb{C}$ be compact and totally disconnected. Is there an elementary way to prove that $\mathbb{C} \setminus E$ is connected?
13
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1answer
4k views

Cofinite topology on an infinite set $X$ is connected?

Here is my proof of that the cofinite topology on an infinite set $X$ is connected. $X$ is connected $\iff$ There are no non-empty disjoint open subsets $U, V \subseteq X$ such that $U \cup V = X$. ...
12
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2answers
1k views

Product of totally disconnected space is totally disconnected?

I read that the cartesian product with the product topology of a family of totally disconnected topological spaces is totally disconnected, too. Is that true? How are the connected components in the ...
12
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2answers
213 views

Connectedness property of $R^2$

My class teacher proposed this problem which seem very interesting. If we remove countably many open disc from $R^2$. Is the remaining space still be path connected. I have done the problem ...
12
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1answer
2k views

Connectedness of points with both rational or irrational coordinates in the plane?

Is the set of points in the plane whose coordinates are either both irrational, or both rational connected?