Questions tagged [connectedness]

Use this tag for question on connected spaces and various related notions (connected components, locally connected spaces, pathwise/arcwise connected spaces, totally disconnected spaces ...) and for connectedness in graph theory.

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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 ...
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  • 7,543
19 votes
4 answers
4k 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.
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  • 3,513
26 votes
10 answers
16k views

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. ...
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37 votes
3 answers
20k views

Topologist's sine curve is not path-connected

Is there a (preferably elementary) proof that the graph of the (discontinuous) function $y$ defined on $[0,1)$ by $$ y(x) =\begin{cases} \sin\left(\dfrac{1}{x}\right) & \mbox{if $0\lt x \lt 1$,}\\\...
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32 votes
5 answers
22k views

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 ...
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42 votes
8 answers
18k 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. ...
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  • 1,968
21 votes
6 answers
21k 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 $A\times\{b\}$ ...
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  • 259
27 votes
6 answers
8k 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 ...
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  • 31.4k
24 votes
2 answers
2k 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 ...
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17 votes
4 answers
8k views

$\mathbb{R}$ \ $\mathbb{Q}$ and $\mathbb{R}^2\setminus\mathbb{Q}^2$ disconnected?

If I want to prove that $\mathbb{R} \setminus \mathbb{Q}$ is disconnected, does it suffice to say that there are two open disjoint sets that cover $\mathbb{R}\setminus\mathbb{Q}$, namely: $$(- \infty,...
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  • 1,861
5 votes
3 answers
2k views

If $h : Y \to X$ is a covering map and $Y$ is connected, then the cardinality of the fiber $h^{-1}(x)$ is independent of $x \in X$.

In "Knots and Primes: An Introduction to Arithmetic Topology", the author uses the following proposition Let $h: Y \to X$ be a covering. For any path $\gamma : [0,1] \to X$ and any $y \in h^{-1}(x) (...
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  • 385
1 vote
2 answers
2k views

Show that $X = \{ (x,y) \in\mathbb{R}^2\mid x \in \mathbb{Q}\text{ or }y \in \mathbb{Q}\}$ is path connected. [duplicate]

How do I show that $X = \left\{ (x,y) \in \mathbb{R}^2 \mid x \in \mathbb{Q}\text{ or }y \in \mathbb{Q}\right\}$ is path connected? Note that $X$ is a topological space with subspace topology $\tau =...
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  • 1,201
14 votes
4 answers
20k views

Proof of "the continuous image of a connected set is connected"

None of the existing questions is exactly answering my question so I'm posting a new question, but feel free to refer me to some already answered question! In Rudin Theorem 4.22, we know that If ...
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  • 2,544
17 votes
1 answer
6k views

What are the components and path components of $\mathbb{R}^{\omega}$ in the product, uniform, and box topologies?

I am working on an exercise problem about components and path components of $\mathbb{R}^{\omega}$. Specifically, Exercise about components and path components: 1. What are the components and path ...
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  • 3,427
4 votes
1 answer
9k views

Intervals are connected and the only connected sets in $\mathbb{R}$

As the topic, prove that Intervals are connected and only connected in $\mathbb{R}$. I know what is the definition of connected set. But not sure how to prove that.
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  • 4,421
28 votes
2 answers
6k 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 $\text{int}(\text{cl}(A)) = A$...
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28 votes
4 answers
28k 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 ...
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  • 2,646
9 votes
2 answers
6k views

Prove that $(X\times Y)\setminus (A\times B)$ is connected

I'm reading topology of Munkres and I have a problem that stuck me for a while. I'm so greatful if anyone can help me with this. Let $A$ be a proper subset of $X$, and let $B$ is a proper subset of ...
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  • 4,475
6 votes
4 answers
3k views

Suppose $Y\subset X$ and $X,Y$ are connected and $A,B$ form separation for $X-Y$ then, prove that $Y\cup A$ and $Y\cup B$ are connected

Question is : Suppose $Y\subset X$ and $X,Y$ are connected and $A,B$ form separation for $X-Y$ then, Prove that $Y\cup A$ and $Y\cup B$ are connected. What i have tried is : Suppose $Y\cup A$ has ...
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21 votes
3 answers
9k 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 ...
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  • 213
8 votes
3 answers
3k views

Formal proof that $\mathbb{R}^{2}\setminus (\mathbb{Q}\times \mathbb{Q}) \subset \mathbb{R}^{2}$ is connected.

Cam anyone provide me the proof of: that $\mathbb{R}^{2}\setminus (\mathbb{Q}\times \mathbb{Q}) \subset \mathbb{R}^{2}$ is connected.
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11 votes
2 answers
7k views

Closed unit interval is connected proof

The closed unit interval $\mathbb{I}=[0,1]$ is a connected subset of $\mathbb{R}$. I am having difficulty understanding the proof in my book, which goes: Suppose that $A,B$ are open sets forming ...
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  • 2,747
531 votes
0 answers
19k 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, $f(S)\subset Y$ ...
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35 votes
2 answers
2k 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 numbers (...
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  • 19.5k
16 votes
3 answers
873 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 ...
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  • 5,730
24 votes
4 answers
14k 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 ...
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  • 885
12 votes
1 answer
3k 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?
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  • 1,002
8 votes
2 answers
6k views

Why does zero derivative imply a function is locally constant?

I've been trying to prove to myself that if $\Omega$ is an open connected set in $\mathbb{R}^n$, then if $f\colon\Omega\to\mathbb{R}^m$ is a differentiable function such that $f'(x)=0$ for all $x\in\...
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  • 1,549
6 votes
2 answers
3k views

Quasicomponents and components in compact Hausdorff space

Let $X$ be a compact Hausdorff space, $x,y\in X$ and $\mathcal{A}$ a colection of closed subspaces of $X$ such that for every $A\in \mathcal{A}$ then $x$ and $y$ are in the same quasicomponent of $A$. ...
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9 votes
1 answer
4k views

Prove that $E$ is disconnected iff there exists two open disjoint sets $A$,$B$ in $X$

Let $(X,d)$ be a metric space. Prove that $E$ is disconnected iff there exists two open disjoint sets $A$,$B$ in $X$ such that $E\cap A\neq\emptyset, E\cap B\neq \emptyset$ and $E\subset A\cup B$. I'...
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61 votes
2 answers
9k 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 ...
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  • 1,047
13 votes
2 answers
11k views

Product of path connected spaces is path connected

Will product of path connected topological spaces be necessarily path connected? Why or why not? Give me some hints. Thank you.
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  • 5,845
10 votes
4 answers
4k views

How to prove this result involving the quotient maps and connectedness?

Given topological spaces $X$ and $Y$, where $Y$ is connected, let $p \colon X \to Y$ be a quotient map. If, for each point $y \in Y$, the set $p^{-1}(\{y\})$ is connected, then how to prove that $X$ ...
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35 votes
4 answers
6k 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}...
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  • 7,622
5 votes
3 answers
1k views

Continuity of function mapping connected set to connected set

If a function maps every connected set onto a connected set, is it necessarily continuous? I know the converse is true.
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5 votes
3 answers
2k views

Prove $\mathbb{R}$ is connected

Prove that $\mathbb{R}$ is connected. PLease i have found other ways to prove it but i want to make this way work. Proof: 1) Strategy : If i show that a arbitrary interval is connected then i can ...
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  • 2,653
8 votes
3 answers
3k views

Family of connected sets, proving union is connected

I am having some trouble trying to prove the following statement:$$$$ Let $(X,d)$ be a metric space and $\mathcal A$ a family of connected sets in $X$ such that for every pair of subsets $A,B \in \...
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  • 3,203
4 votes
2 answers
4k 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 $X$. ...
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  • 1,796
32 votes
3 answers
2k 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-...
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  • 341
40 votes
6 answers
2k 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&...
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  • 46.5k
28 votes
6 answers
1k views

Could *I* have come up with the definition of Compactness (and Connectedness)?

Ok, buckle up for a rather long question. I've spent a large portion of today learning about compactness, stemming mainly from this wikipedia article about point-set topology. The article mentions ...
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  • 5,896
5 votes
2 answers
4k views

Connected topological spaces, product is connected

Show that if $(X_i)_{i \in \mathcal I}$ where $X_i$ is a topological space for every $i \in \mathcal I$, then $X_i$ is connected for every $i$ if and only if $\prod_{i \in \mathcal I} X_i$ is ...
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  • 3,629
13 votes
1 answer
809 views

"Antisymmetry" among cut points

We might define a ternary relation between points of a topological space $X$ by writing $x|yz$ whenever $y$ is not in the same quasi-component of $X\setminus \{x\}$ as $z$. It is not hard to prove ...
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11 votes
2 answers
7k views

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

I'm starting with the basics of topology theory and I'm trying to show: Let $A$ and $B$ be closed subsets of a topological space. If $A\cap B$ and $A\cup B$ are both connected, $A$ and $B$ are ...
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  • 115
16 votes
4 answers
4k views

Continuous image of a locally connected space which is not locally connected

The question is pretty much in the title, I'm looking for an example of a locally connected space and continuous mapping such that the image is not locally connected. Thanks! EDIT: Corrected the ...
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  • 3,535
14 votes
1 answer
559 views

Is there an infinite connected topological space such that every space obtained by removing one point from it is totally disconnected?

The particular point topology on any set is connected, but on removing the particular point, the complement is discrete, and hence totally disconnected. Although this is not even $T^1$, Cantor's leaky ...
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  • 6,423
8 votes
5 answers
12k views

Closure of a connected subset of $\mathbb{R}$ is connected?

Prove or disprove: The closure of a connected set in $\mathbb{R}$ is always connected. Response: I don't really have a grasp on this conceptually, but here's an attempt. Proof: Let $X$ be a connected ...
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  • 583
17 votes
3 answers
2k views

Are there any countable Hausdorff connected spaces?

Do countable Hausdorff connected topological spaces exist?
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  • 1,359
10 votes
3 answers
1k views

Looking for a counter example for non-connected intersection of descending chain of closed connected sets

Let $X$ be a topological space and let $\left\{ Y_{i}\right\} _{i=1}^{\infty}$ be a descending chain of closed connected subsets of $X$. I know from reading elsewhere that ${\displaystyle \bigcap_{i=...
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  • 3,535
7 votes
2 answers
4k views

If $X$ and $Y$ are connected, then $(X\times Y)\setminus(A\times B)$ is connected for any proper subsets $A,B$

I meet these two exercises: Q1: let $A$ be a proper subset of $X$, and $B$ be a proper subset of $Y$. If $X$ and $Y$ are connected, show that $(X\times Y)\setminus(A\times B)$ is connected. Q2: ...
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