# Tagged Questions

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### 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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### Openness of path connected components of open subsets of $\mathbb C$

Let $\Omega\subset \Bbb{C}$ be an open set. My textbook states that every path connected component of $\Omega$ is open. I can't seem to understand why that is. Why does every point have to contained ...
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### The intersection of a connected subspace with the boundary of another subset

Can someone please verify my proof or offer suggestions for improvement? Definition/Notation: The boundary of $A$, denoted by $\operatorname{Bd}(A)$, equals $\overline{A} \cap \overline{X-A}$. ...
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### Connectedness and non-local-connectedness of a subspace of $\mathbb R^2$

Let $(X,\tau)$ be the subspace of $\mathbb R^2$ consisting of the points in the line segments joining $(0,1)$ to $(0,0)$ and to all the points $(1/n,0)$, $n=1,2,3,\ldots$. Show that $(X,\tau)$ ...
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### Cardinality of fibres of covering maps of connected spaces

If I have a covering map $p:E \rightarrow B$ and some connected set $U$, that is evenly covered, then $p^{-1}(U)$ as a partition into slices is unique. Now, if I assume that $B$ is connected, then I ...
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### Connected subsets of metric (or T1) spaces

I have proved some statements about connected subsets of a metric space. They are really basic, but I want to make sure that they are true. Would someone please tell me whether these statements are ...
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### Connectedness of both $Y \cup A$ and $Y \cup B$ where $A, B$ is a separation of $X -Y$

Let $Y\subset X$ be such that both $X$ and $Y$ are connected. Show that if $A$ and $B$ is a separation of $X-Y$, then $Y\cup A$ and $Y\cup B$ are connected. I found a proof for this problem in this ...
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### Path Connectedness argument for $SO(n, \mathbb{R})$

I am trying to prove path connectedness of $SO(n, \mathbb{R})$. I have seen several different proofs for the same. But I had a thought and wanted to know whether it would help in any way. I took two ...
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### Cover a sphere by two closed subsets not containing a closed self-antipodal connected subset?

Question (Fulton's Algebraic Topology, A First Course, Problem 4.40) Suppose the sphere $S^2=A\cup B$ where $A,B\subseteq S^2$ are two closed subsets of $S^2$. Is it true that either $A$ or $B$ must ...
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### Connected Sets in Topology

Theorem: Let $(X,\mathscr{T})$ be a topological space. If $E$ is connected and $K$ is such that $E\subseteq K\subseteq\mathrm{cl}(E)$, then $K$ is connected. (Cl(E) is closure of E) Question: ...
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### Is the sphere $S^n$ always arcwise connected?

I have a small question about the connectedness of the sphere; Is the sphere $S^n$ always arcwise connected ? Thank you.
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### Simple question on connectedness in a subspace [duplicate]

For some reason I am having some trouble on this basic point set topology question: Suppose $X$ is connected, and $A$ is a connected subset of $X$, and that $B$ is a clopen set in $X-A$ (not in $X$, ...
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### Proof: X is connected

Just came from an exam and I am wondering how to prove the following: A topological space $X$ is connected if for each continuous function $f:X\rightarrow X$ there is a $x \in X$ such that ...
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### Connectedness and some properties [closed]

Let $S \subset \mathbb R^2$ be defined by $$S = \left\{ \left(m + \frac{1}{2^{|p|}}, n + \frac{1}{2^{|q|}}\right) : m,n,p,q \in \mathbb Z \right\}.$$ Is $S$ discrete? Is $\mathbb R^2\backslash S$ ...
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### Prove that intersection of connected spaces is connceted.

Let A and B be connected subspaces of a topological space (X,$\tau$). If A,B are not disjoint, prove that the subspace A $\cap$ B is connected. Using the definition of connected space is that the ...
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### Step Connected if and only if Connected

A space $X$ is step connected if given any open covering $\mathcal{U}$ of $X$ and any pair of points $p,q\in X$ there is a finite sequence $U_1,\ldots,U_n$ of sets belonging to $\mathcal{U}$ so that ...
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### A maximal subset of $S^2$ with respect to a connectedness property

Let the set $A$ be a circle with a chord on the sphere $S^2$. Obviously $A$ has the following property: P: $\quad$ Any two points $a$ and $b$ of $A$ can be connected by a path that ...
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### Disconnecting using totally disconnected sets [duplicate]

Let $X$ be $[0,1]^2$ and $S\subset X$ a totally disconnected subset. Is it true that $S^c$ is always connected? If it is false, what can we say when $X=[0,1]^n$?
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### A question of topology.

If S is a subset of $\hspace{0.1cm}$$[0,1]\times[0,1]$$\hspace{0.1cm}$ such taht one point of the ordered pair is rational and the other is irrational or both are irrationals,then which of the ...
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### Is for open connected $U$ the set $U_\varepsilon$ for small $\varepsilon$ connected?

Let for an open connected subset $U\subset \mathbb R^n$ and a number $\varepsilon >0$: $$U_\varepsilon=\{x\in U: dist (x, \partial U)> \varepsilon \}.$$ Then $U_\varepsilon$ is open but in ...
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### Closed and Connected subgroups of $\mathbb{R}^n$

Question is : What are closed connected subgroups of $\mathbb{R}$ and from that deduce what are closed connected subgroups of $\mathbb{R}^n$ What i have done so far is : Only connected subsets of ...
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### Product of Connected Spaces (2)

If $Y$ and $Z$ are connected, is $Y \times Z$ path connected? I cannot find a counter example. Some help please.
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### When does connectedness imply path-connectedness

In a locally path-connected space connectedness and path connectedness are equivalent. What is the minimal condition we would impose on a topological space to get the same result?
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### $A\cup B$ is connected when $A$ is connected in $X$ and $B$ clopen in $X-A$

Let $A$ be connected subset of a connected space $X$, and $B\subset X-A$ be an open and closed set in the topology of the subspace $X-A$ of the space $X$. Prove that $A\cup B$ is connected. I think ...
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### Prove: If H and G/H are totally disconnected then G is also totally disconnected

Let $G$ be a topological group and $H$ a subgroup of $G$. If $H$ and $G/H$ are totally disconnected, then $G$ is also totally disconnected. With 'totally disconnected' we mean the every connected ...
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### Finding the connected components of topological spaces

Find the connected components of the following sets: $(a) \; A=\{(x,y):y=\sin(1/x), x\in\mathbb{R}^+\},(b)\;A\cup\{(x,y):x=0,y\in[-1,1]\},(c)\;$The Cantor Set,$(d)\;\mathbb{N}$ with the cofinite ...
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### Deleting $n$ points from a connected space

Let $X$ be a space such that for any subset $S \subset X$ with finite cardinality $n$, the subspace $X \setminus S$ has exactly $n+1$ connected components, each of which is homeomorphic to $X$. Is ...
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### Möbius band with its middle part removed is still connected

Let $I\times I/(0,t){\sim}(1,1-t)$ be the Möbius band and let $S=\{(x,y): (x,y)\in M, 1/4<y<3/4\}$ be its middle part. How can I show that $M-S$ is connected? I tried to write a continuous ...
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### Can a connected subpsace be in disjoint open sets?

Say $X$ is a non-connected topological space, i.e. $X= U\cup V$, and $U,V$ are disjoint (non-empty) open sets. Then suppose $C$ is a connected subspace of $X$, with the standard subspace topology. Can ...
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### Connectedness problem on the 2-sphere

Let $K \subset L \subset S^2$, where $S^2$ is the 2-sphere and $K$ and $L$ are compact subsets with empty interior and $L$ is connected (I don't think all of those are relevant hypotheses though, but ...
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### Examples of continuous integer-valued functions on totally disconnected spaces

I wanted to see examples of continuous integer-valued functions $f:X\to \mathbb{R}$ on a totally disconnected space $X.$ I have only some abstract examples in mind.
Let $\mathbb{J} :=\{1/n: 0< n\in \mathbb{Z}\}$ Let $T_{ir}$ be topology of $\mathbb{R}$ generated by \{(a,b)\subset \mathbb{R}:a<b\}\cup\{(a,b) \setminus \mathbb{J}\subset ...