# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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: ...