# 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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### On the completeness of Weak Operator Topology

Let E,F be any two Banach space and let $\mathcal{B}(E,F)$ be the space of all bounded linear operators from E to F. I can show that this space is a complete space with respect to the norm and strong ...
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### Which surface is homotopy equivalent to $\Bbb{R}^4$ minus the planes $x=y=0$, $z=w=0$?

In completing an exercise I have shown that $\Bbb{R}^3$ minus the axes $x=0$, $y=0$, and $z=0$ is homotopic equivalent to the cube graph $Q_3$. To visualize this, $\Bbb{R}^3-0$ is homotopy equivalent ...
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### What is the $\epsilon$ neighborhood of a subset in $\mathbb R^2$ in $\mathbb R^n$

Let X denote the subset $(-1,1) \times 0$ of $\mathbb R^2$ and let U be the open ball B(0,1) in $\mathbb R^2$ which contains X. Show that there is no $\epsilon > 0$ s.t the $\epsilon$-neighborhood ...
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### Showing $D = \{ f : |f(x) - f(y)| \leq \sqrt{|x-y|} \}$ where $D \subseteq C[0,1]$ is not bounded

Given $D = \{f: |f(x) - f(y)| \leq \sqrt{|x-y|}\}$ where $D \subseteq C[0,1]$, I am asked to show this set is not compact. To do so, I must show the set is not bounded (specifically not uniformly ...
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### Show that minimum and maximum are contained in $V(\mathcal{R})$/ Stones' axiom

Let $\mathcal{R}\subset\mathcal{P}(\Omega)$ be a ring for some set $\Omega$. Consider  V:=V(\mathcal{R}):=\left\{\sum_{i=1}^n\alpha_i1_{A_i}: \alpha_i\in\mathbb{R}, A_i\in\mathcal{R}, ...
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### CW construction of Lens spaces Hatcher

I am working through Hatcher's book and I am having trouble while understanding the CW-complex structure of Lens spaces. It is on page 145. He proves it constructing it in an inductive process. I ...
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### the point set is nowhere dense in $X$

I am trying to show for a metric space $X$, a set $\{x\}$ consisting of a single point is nowhere dense. I have proven it by showing $[{\{x\}}]^o = \emptyset$ where $[A]$ is the closure of the set $A$ ...
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### $T_0$ is equivalent to $T_1$ in a topological group

I need to prove that the separation axiom $T_0$ is equivalent to $T_1$ in a topological group $G$. $T_1$ implies $T_0$. So I only need to prove $T_0$ inplies $T_1$. What I have tried is: ...
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### Fundamental Groups of a Simplicial Complex and the Underlying Space

Now I know how the fundamental group of a Simplicial Complex is defined, as well as that of a Toplogical Space. Could someone explain the process of how we prove that for a simplicial complex $X$ , ...
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### 2-Cocycles and Automorphisms

hopefully a short question, which can be answered by someone who is more into that topic than I am. Suppose we have a $\mathbb{Z}_2$ valued 2-cocycle $\epsilon$ defined on a root lattice $\Phi$ of a ...
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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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### Why this doesn't determine a torus?

Why the diagram on Figure 3.6 doesn't determine a torus? Do we need at least three rows of rectangles?
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### Canonical metric on the suspension of a metric space

Let $(X,d)$ be a metric space. Is there a metric $d'$ on the (unreduced) suspension $\Sigma X = (X\times[-1,1])/\sim$ of $X$ such that $d'$ restricts to $d$ on $X\times \{1/2\}$? Further, we would ...
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### Isolated and discrete points

In the space of integers with usual topology, is it true that 1) every point is isolated point for any set 2) no point is limit point for any set. In real space and space of rationals 1)no point is ...
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### Example of dense set in space of integers

Let X be the space of integers where topology T is class of sets $\emptyset, \{1\}, \{1,2\}, \{1,2,3\},\{1,2,3,4\} \cdots, X$. Is it correct that each open set except $\emptyset$ is dense in space of ...
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### Closure of a set in topological space

Closure of a set $A$ in topological space $X$ is defined as intersection of all closed supersets of $A$. Can we say that $Closure(A)= \bar{A}$ such that $(\bar{A})'=\bigcup B$ where $B$ is the class ...
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### Let $(x_\alpha)_{\alpha\in J}$ be a net in a topological space $X$. Show that if $J$ is a ﬁnite set then the net converges? …

Let $(x_\alpha)_{\alpha\in J}$ be a net in a topological space $X$. Show that if $J$ is a ﬁnite set then the net converges? ... Why does it have to converge? I don't understand why with nets this is ...
Let $(X,x_0)$ be a $H$-space with multiplication $\mu:X\times X\to X$. Let $e$ denote the constant map $I^n\to x_0$. Is it true (and why) that \$\begin{cases} ...