# 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.

1,906 questions
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### 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 ...
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### 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 ...
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### 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). ...
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### $(A_i)_{i\in E}$ 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 ...
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### 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 $X$...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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....
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### 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 ...
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### 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) ...
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### 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 ...