# Tagged Questions

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; ...

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### If a product space is locally compact, then each space is locally compact and all but a finite number of factors are compact

If $\prod^{\infty}_{i=1} (X_i, T_i)$ is locally compact, then each $(X_i, T_i)$ is locally compact and all but a finite number of $(X_i, T_i)$ are compact. Let $X=\prod^{\infty}_{i=1} (X_i, T_i)$, ...
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### Classification of an open set in real

Prove that open set in real line can be represented as ar most countable disjoint union of open intervals. I know that this question repeated many times in MSE but let me ask the following question. ...
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### Determine the closure, interior and boundary of the set

What does it mean when it asks for the interior of the set? Also to check, I think this set is open with the boundary at x=0 and an open disk with a radius 1. Am I correct?
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### Urysohn's Lemma from RCA Rudin

I found out the proof of Urysohn's Lemma from Rudin's book but I have couple questions which I am not able to answer. 1) Why Rudin wrote that "in terms of characteristic functions, the conclusion ...
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### Subfields of $\mathbb{C}$ which are connected with induced topology

The ring of continuous functions on $[0,1]$ to $\mathbb{R}$ has an interesting property: every maximal ideal of this ring is the subset of all functions vanishing at a common point. If we ...
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### Is $Y/K$ homeomorphic to $Y'$ as defined below -

Let $G$ be a topological group acting on a topological space $X$ in such a way that there are only finitely many orbits. We will fix points $x_1,\cdots,x_n\in X$ and let $X=\bigcup_{i=1}^n G\cdot x_i$ ...
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### can we find a continuous surjection from $\mathbb{R} \to \mathbb{R}^{\omega}$?

I've shown there exist continuous onto map from $\mathbb{R}$ to $\mathbb{R}^{n}$ for any finite $n$. Now my question can we find a continuous surjection from $\mathbb{R} \to \mathbb{R}^{\omega}$ ? ...
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### Is the complement of the closed unit disk in the plane homeomorphic with $\mathbb R^2\setminus \{(0,0)\}$ ? [closed]

Is $\mathbb R^2 \setminus D^2$ , where $D^2=B[0;1]$ is the closed unit disk , homeomorphic with $\mathbb R^2\setminus \{(0,0)\}$ ?
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### Prove Theorem 2.30 from baby Rudin.

I want to prove that Suppose $Y\subset X$. A subset $E$ of $Y$ is open relative to $Y$ if and only if $E=Y\cap G$ for some open subset $G$ of $X$. The following statement is a bit confusing to ...
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### Is the plane minus a line segment homeomorphic with punctured plane?

Is $\mathbb R^2$ minus a line segment i.e. $\mathbb R^2 \setminus ([0,1]\times \{0\})$ homeomorphic with a punctured plane $\mathbb R^2\setminus \{(0,0)\}$ ?
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### Generators of the fundamental groups of the 8-figure and the torus

I have two doubts strictly related to each other. 1) Firstly, consider the $8$-figure, namely the union of two circles in a point $x_1$. Using the Seifert-Van Kampen's theorem I proved that its ...
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I have been reading Massey's Algebraic Topology and on page 158 came across the following "semi-mystical principle" which he says guides much mathematical research: Whenever we wish to gain ...
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### Inverse function on product topology (Munkres)

I have a simple question that comes from Munkres section 19, Example 2. Let $f:\mathbb{R}\rightarrow\mathbb{R}^{\omega}$ be given by $f(x)=(x,x,x,...)$, with $\mathbb{R}^{\omega}$ a countably ...
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### Proof verification: diam(E) = diam(closure(E))

Since $E \subseteq cl(E)$, then it is immediate that diam $(E) \leq$ diam(cl($E))$. I only need to show that assuming diam $(E) <$ diam(cl($E))$ will lead to contradiction then I can conclude ...
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### Can someone please offer a simple definition of “derived net”?

I was looking up a term called "derived net", however the Google search seems to conflate "net" with .NET programming language. (And filter with electronic filters, and "derived net from a filter" ...
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### Show that if $x$ is an accumulation point of an ultrafilter on $X$, then the neighborhood filter is contained in the filter

Show that if $x$ is an accumulation point of an ultrafilter $\mathcal{F}$ on $X$, then the neighborhood filter $\mathcal{F}_x$ is contained in the filter i.e. $\mathcal{F}_x \subseteq \mathcal{F}$ ...
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### Characterizing spaces with no nontrivial covers

I know that simply connected locally path-connected spaces have no nontrivial covers. Is there a characterization of spaces with this property?
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### In $T1$ space, all singleton sets are closed?

The definition of $T1$-Space is: A topological space $X$ is said to be $T1$ if for each pair of distinct points $a,b,$ $\exists$ open sets $U,V$ s.t $a\in U, b\notin U, a\notin V, b\in V$. What ...
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### Random set of rationals topological properties

Flip a coin (probability of heads is p, strictly greater than 0 and strictly less than 1) for every rational number. For each toss, if heads include the number in a set S, if tails exclude it. What is ...
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### How can a countably infinite product space be discrete?

Let $(X_i,T_i), i \in \Bbb N$, be a countably infinite family of topological spaces. Prove that $\prod^{\infty}_{i=1} (X_i,T_i)$ is a discrete space iff each $(X_i, T_1)$ is discrete and all but a ...
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### Show that $\mathbb{Q}$ with the topology induced by the Sorgenfrey line is normal

Show that $\mathbb{Q}$ with the topology induced by the Sorgenfrey line is normal. Here, the Sorgenfrey line is the line with the topology whose basis are formed as: $$\{[a,b) : a < b \}$$ I ...
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### On the matter ; If $f:X \to Y$ is a function with closed graph and compactness preserving then $f$ is continuous

Let $X,Y$ be metric spaces , $f:X \to Y$ be a function , with closed graph , carrying compact sets to compact sets ; then I claim that $f$ is continuous Proof: Let , if possible , $f$ be not ...
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### Infimum of lower semicontinuous functions

The following proposition is from the book Nicolae Dinculeanu Integration on Locally Compact Spaces: Let $H$ and $K$ be two compact Hausdorff spaces and $\alpha$ a continuous mapping of $H$ onto $K$. ...
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### (Real Analysis) Topology: Prove $f(cl S)\subseteq clf(S)$

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be continuous. Show: $f(\overline{S})\subseteq \overline{f(S)}$ for $S\subseteq \mathbb{R}$ (Note: $\overline{S}$ denotes the closure of S; $\partial S$ ...
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### Hausdorff & locally compact spaces

1) Every compact space is locally compact. 2) Every metric space is a Hausdorff space. 3) $\mathbb{R}^n$ is a locally compact space. Proof: 1) Suppose $X$ - compact space. Taking $p\in X$ we see ...
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### Confusing moment from Theorem 2.7 Rudin RCA

I am working with theorem 2.7 from Rudin's RCA book and one moment worries me. Here Rudin uses theorem 2.5 which I added above. Using theorem 2.5 we get that $p \notin W_p$. But how he concludes ...
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### (Conceptual) Continuity of binary relation $\succsim$ and definition using contour sets

Some background information: $\succsim$ is a binary relation that represents preference between two goods. $\succsim$ means "x is at least as good as y." Continuity of this relation is defined to be ...
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### If each $(X_i,T_i)$ is finite and discrete, then the box product is not compact.

If each $(X_i,T_i)$ is finite and discrete, then the box product is not compact. Let each $(X_i,T_i) = \{x_i\}$, then the box product is only covered by the open set $U = \prod \{x_i\}$, but since ...
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### An element is in the cantor set iff it can be written in ternary form with $a_n \neq 1$, for all $n \in \Bbb N$

An element of $[0,1]$ is in the cantor set iff it can be written in ternary form (base $3$) $(0.a_1 a_2 ... a_n ...)$ with $a_n \neq 1$, for all $n \in \Bbb N$. How is this possible? The book I'm ...
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### Product of Ultrafilters: How to show $p,q <_{RK} p\otimes q$?
Let $p,q$ be two free ultrafilter on $\mathbb{N}$, i.e. elements of $\mathbb{N}^* = \beta\mathbb{N}\setminus\mathbb{N}$. The Rudin-Keisler order is defined as follows: $p \leq_{RK} q$ iff there is a ...