# 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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### Product of connected sets is connected

I know that this question had already been asked here but there is a problem ... all the proofs that I've seen used homeomorphisms and continuous functions. Well my teacher didn't teach me what a ...
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### When does path connectedness implies convexity?

It is easy to see that every convex set is path connected. What are some examples so that converse holds (not counting the (trivial) one dimensional case)? Is there a nice topology so that this holds? ...
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### equivalence of arakelian set definition

A closed set $E$ without holes ( holes means bounded component of its complement) is said to be Arakelian if for every disc $D$ ( or compact sets in general) the union of all holes of $E \cup D$ ...
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### Connected components of an open subset

Suppose we have an (affine) irreducible algebraic variety $X$ over $\mathbb{R}$, and we are given one polynomial $f$. Consider the closed subset $C$ where this is zero. What can we say about the ...
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### Prob. 6, Sec. 27, in Munkres' TOPOLOGY, 2nd ed: The Cantor's set is totally disconnected, compact, and uncountable

Here is Prob. 6, Sec. 27, in the book Topology by James R. Munkres, 2nd edition: Let $A_0$ be the closed interval $[0, 1]$ in $\mathbb{R}$. Let $A_1$ be the set obtained from $A_0$ by deleting its "...
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### Probs. 3 (a & c), Sec. 27, in Munkres' TOPOLOGY, 2nd ed: The space $\mathbb{R}_K$ is not path connected and $[0, 1]$ is not compact in $\mathbb{R}_K$.

Here is Prob. 3, Sec. 27, in the book Topology by James R. Munkres, 2nd edition: Recall that $\mathbb{R}_K$ denotes $\mathbb{R}$ in the $K$-topology. (a) Show that $[0, 1]$ is not compact as a ...
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### How is a $k$-form integrated over an oriented smooth $n$-manifold in the case it is connected?

I have seen in several answers to questions on this page stating that there is no way to integrate a $k$-form over an oriented smooth $n$-manifold if $k \neq n$. However I cite Tu in his book on ...
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### Is it possible that there is a connected topological space without path-connected subspace?

Is it possible that there is a connected topological space without path-connected subspace? Furtherly ~ Is that any connected topological space $X$ always has dense path-connected subspace? Or Is ...
I have this theorem. let $A$ and $B$ two closed sets of a topological space $E$, such that $A\cap B$ and $A\cup B$ are connected, then $A$ and $B$ are connected. I want to prove it directly ...
### Show that a smooth map $F : M \rightarrow N$ has Constant rank if $F$ has a linear coordinate representation.
Suppose $F:M\to N$ is a smooth map, with $M,N$ smooth manifolds and $M$ connected. I want to show that if for each $p$ in $M$ there exists smooth charts near $p$ and $F(p)$ in which the coordinate ...