# Tagged Questions

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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### If $A$ is a component in $(X,d_1) \iff$ $A$ is a component in $(X,d_2)$ then is $(X,d_1)$ homeomorphic to $(X,d_2)$?

Suppose that $A$ is a component in $(X,d_1) \iff$ $A$ is a component in $(X,d_2)$. Is it always the case that $(X,d_1)$ is homeomorphic to $(X,d_2)$? I have been trying to find a counter example, but ...
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### Why is a circle not simply-connected?

To be simply-connected means to be path-connected and able to continuously shrink a closed curve while remaining in the domain. According to wikipedia, a circle is not simply connected, but a disk ...
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### Find a connected graph that has exactly $2$ cutpoints of order $2$ and $3$ cutpoints of order $3$

Find a connected graph that has exactly $2$ cutpoints of order $2$ and $2$ cutpoints of order $3$ Definition: A cut point of order $k$ is a point $a \in X$ whose complement $X-\{a\}$ consists of ...
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### Is this proof regarding product of connected spaces correct?

Let $X,Y$ be connected spaces, and consider their product $X\times Y$. I want to show that their product is connected. The posts I've read here regarding this question often include creating "slices" ...
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### Why are the (connected) components of a topological space themselves connected?

I am trying to prove that (connected) components of a topological space are connected. I'll first define what I mean by a 'component of a topological space': For a topological space $X$, write ...
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### Showing the Following Set is Path Connected

Would be grateful if someone could help me with the following question "For any normed space X which is NOT the set of Reals, the set X (excluding 0) is path connected In the solutions the usual ...
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### Union of the unit circle $S^1$ and the curve is connected but not path-connected

Prove that the union of the unit circle $S^1$ and the curve $W=\{(x, y) \in \mathbb{R^2} | x=(1-e^{-t})cost, y=(1-e^{-t})sint, t \geq 0\}$ is connected but not path-connected A connected space is ...
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### If $\Omega$ is an open set of $\mathbb{C}$, $f$ constant on each connected component then $f$ is continuous

Let $\Omega$ be an open set of $\mathbb{C}$ and $f$ a constant function on each connected component of $\Omega$. I need to proof that $f$ is continuous. I've tried using that connected components ...
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### Is it possible to prove this using set theory only and no more?

Got to prove: If $E$ and $F$ are connected sets, and $A$,$B$ are subsets of $E$ and $F$ respectively (but neither $A$ or $B$ are empty or fill $E$ and $F$entirely). Then $(A\times B)^c$ is connected ...
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### An equivalent definition of connectedness in General Toplogy

I'm studying General Topology by the book of James Munkres and there he defined connectedness this way: "Let X be a topological space. A separation of X is a pair U, V of disjoint nonempty open ...