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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Topology of $L^2$ space

Cardinality of space of all funcions $f: \mathbb R \rightarrow \mathbb R$ is $\beth_2$. However, cardinality of space of all such square-integrable functions, space $L^2$, is $\beth_1=\mathfrak c$, ...
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38 views

Modes of convergence for continuous functions

I just wondered about what modes of convergence for continuous functions $f_n:X\rightarrow Y$ between topological spaces there are. Of course there is pointwise convergence, which is defineable for ...
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73 views

Convention of a continued fraction presentation of a lens space

I want to clarify the following two conventions on a surgery description of a lens space. Let $p$ and $q$ are relatively prime integers. Express $$ p/q=x_1-\cfrac{1}{x_2-{\cfrac{1}{x_3-\dots\cfrac{1}{...
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42 views

Contractibility of complex manifold

I'm trying to show that for $v^2 = w^4 - a^4$ for real $a$ and complex $v, w$ that this manifold deform retracts to a point $(0, a)$ but can't seem to figure out a path that remains on the surface. ...
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37 views

Retraction and intersection

Let $X$ be a topological space, and consider two open subsets $U$, $V$ of $X$ such that there exist two continuous maps $r_{U}: X\longrightarrow U$, $r_{V}:X\longrightarrow V$ which are homotopically ...
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50 views

Continuity of the dual product reloaded

Let $X$ be a Banach space with topological dual $X^*$. Then the dual product $(x,x^*)\in X\times X^*\to x^*(x)\in\mathbb{R}$ is strongly$\times$strongly continuous in $X\times X^*$. That does not ...
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47 views

The compactification $\hat{E}_{\mathcal{F}}$ is the Alexandroff-comp. if $E$ is discrete and $\mathcal{F}$ only contains constant functions

this is my task: Prove that if $E$ is a discrete set and if $\mathcal{F}$ contains only constant functions then the compactification $\hat{E}_{\mathcal{F}}$ of $E$ with respect to $\mathcal{F}...
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48 views

Orbit space of action of a subgroup of a Lie group on a separable metric space

I am stuck on this question. Let $G$ be a Lie group acting freely on a separable metric space $X$. Assume that the orbit space $X/G$ is Hausdorff. Let $H$ be a normal Lie subgroup of $G$. Is the orbit ...
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45 views

Find a topology $\mathscr{T}_0$ for which $f,g : ( \mathbb{R}^2, \mathscr{T}_0) → (\mathbb{R}^2, \mathscr{T}_{R}^{c} )$ is continuous.

For $f,g : \mathbb R \rightarrow \mathbb R$, define: $$ x \rightarrow f(x) := \begin{cases} x^2 & \text{for }x \le 1 \\ x+1 & \text{for }x > 1 \end{cases} $$ $$ x \rightarrow g(x) := \...
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34 views

Borel sigma algebra and an extra point

I have a question about Borel sigma algebra on a topological space. Let $E$ be a Hausdorff topological space and $\mathcal{B}(E)$ denotes its Borel sigma algebra. We adjoin an extra point $\Delta$ ...
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41 views

Weak /Strong operator topology?

Could someone explain me what weak and strong topologies are and provide some practical example of their use in, for instance, feature checking in data computation and in the study of movement in ...
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33 views

Is there a complete Link Invariant for links with N crossing.

Are there known examples of pairs $\left(f, N\right)$, where $f$ is a link invariant that is known to be complete when restricted to link diagrams that have at most $N$ crossings? (Ideally, f should ...
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84 views

Converging sequence implies limit point

Is it true that if a sequence in a metric space converges to a value, then that value is a limit point of the set of all terms in the sequence? $E = \{ p_1, p_2, \dots, p_n , \dots \} \subset X$, ...
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50 views

Space generated by a reflection

Suppose I embed a mirror (not necessarily plane) in some space (say a manifold). Is there a theory that tells you how to classify the "space" generated by the reflection (the one you see if you were ...
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59 views

How to prove a function is harmonic polynomial

1! How to prove this function a harmonic polynomial using Laplace equation For the 1 question I know we can prove harmonic using Laplace Equation but for this on m confused how to start. For the 2 ...
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51 views

Is the question in the Munkres's topology book wrong?

At the end of cheapter $8.1$, $4)$ Given spaces $X$ and $Y$, let $[X,Y]$ denote the set of homotopy classes of maps of $X$ into $Y$. $b)$ Show that if $Y$ is path connected, the set $[I,Y]$ has a ...
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270 views

Every point in the open set is a limit point

I know that closed set contains all of its limit points. However, I can claim a statement: Every point in the open set $O$ is a limit point. Here is my proof: Suppose $x \in O$. By definition, $\...
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33 views

Existence of nets from weak closure

Hi I am interested in the following question. Given some normed space $X$ with a subset $S \subset X$. If I consider $x \in \text{wcl}(S)$, where 'wcl' denotes the weak closure of $S$, then since the ...
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94 views

A contradiction of product topology

Let $X$ and $Y$ be topological spaces and $X\times Y$ be their product. The product topology on $X\times Y$ is the topology generated by the basis $B = \{U\times V | U$ is open in $X$ and $V$ is ...
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46 views

Can there be weak* open cover of the dual banach space with the arbitrary small (in diameter) sets?

That is, I want to cover $X^*$ (X is Banach space) with a family $\{U_{\alpha}\}$, where $diam(U_{\alpha})<\epsilon$ and each $U_{\alpha}$ is weak* open. I expect, that not every open ball is weak*...
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58 views

Number of topologies on a set

Let $X$ be a nonempty set with $n$ elements. I want to find an upper bound for the number of possible topologies for $X$. I proceed as follows: The power set $\mathcal P(X)$ contains $2^n$ elements. ...
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67 views

A counter example

If a set is compact in $Z(\mathbb{A})\setminus GL(2,\mathbb{A})$,then can it be compact in $GL(2,\mathbb{A})$ ? ($\mathbb{A}$, is the adele ring of $F$ on which $GL(2)$ is and $Z$ is the center of $GL(...
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59 views

Importance of metrization theorem?

I wonder if there is a case metrization theorems(such as Nagata-Smirnov, Bing, Urysohn) pave a way to do a theory. What would be a nice application of metrization theorems?
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76 views

$E$ is compact if and only if X is compact and $q$ is a finite-sheeted covering

Let $q:E\to X$ be a covering map. Then $E$ is compact if and only if X is compact and $q$ is a finite-sheeted covering. My question is regarding the $"\implies"$ direction: If $E$ is compact, then $X$...
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59 views

If $X$ and $Y$ are homotopic and $X$ is contractible, so is $Y$

I want to show that if $X$ and $Y$ are homotopic and $X$ is contractible, so is $Y$. It feels like I'm missing something really obvious. $X$ is homotopic to $Y$, so there exists $f: X \to Y$ and $g: Y ...
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63 views

Number of connected components of $f^{-1} (U)$

Let $f:\mathbb{R}^n \to \mathbb{R}$ be an analytical function (semialgebraic,polynomial if needed), $U$ be an open connected subset of $\mathbb{R}$. What can we say about the nuber of connected ...
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149 views

Can you deformation retract a sphere to a point?

So, I'm working on a topology problem (Calculating the fundemental group of two spheres adjoined by a single point). As a subpart of the problem, we're trying to figure out if a sphere by itself can ...
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55 views

Proof of Version of Krasner's Lemma

I need some help in constructing the proof to this version of Krasner's Lemma from Serre's Local fields text book: Let E/K be a finite Galois extension of a complete field K. Prolong the valuation ...
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Degrees of spaces of polynomials

Let $I$ be an ideal in $K[x_1,\dots,x_n]$ where $K$ is a char $0$ field. Let $Z(I)$ be a set of discrete points whose cardinality is exponential in $n$ and spanning $n$ dimensions. Let $P$ be the ...
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57 views

What's the general technique to show a sequence converges?

After "guessing" what the limit of a particular sequence is, what's the general process to prove that this sequence indeed converges to it? (using the definition) (The definition says that a sequence ...
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Proof of Theroem 2-7 in Rudin's Real and Complex Analysis

I am working on the proof of Theorem 2-7 in Rudin's Real and Complex Analysis and I need some help clarifying why p does not belong to the closure of Wp. Here is a link to the entire theorem. ...
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78 views

Prove that this space is contractible

Consider the space $X/\sim=((-1,1)\times \{0,1\})/\sim$ where $(x,0) \sim (x,1)$ for $x\leq0.$ Prove that $X/\sim$ is contractible. I suppose we can contract $X/\sim$ to the point $[(0,0)] = [(0,1)]$....
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What does $R^I$ stand for?

In section 30 of Munkres, one exercise states that "Give $R^I$ the uniform metric, where $I=[0,1]$". I guess it's not about powers or something, it's some conventional notation because I've never ...
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92 views

Lindelöf Property and Compact space

Let $X$ be a compact space and $L$ is the smallest family of subspaces of$\,X\,$that contains all closed sets and is closed with respect to countable union and intersection. The question is :- Is ...
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71 views

Levels sets of a continuous function

Suppose $f:[0,1]\rightarrow [0,1]$ is continuous. Let $A$ be the set of all maximal, connected subsets of the level set $f^{-1}(0)$. Can $A$ be uncountable?
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62 views

How is this topological space different from the euclidean one?

I'm preparing for my topology exam and came across this example which I can't figure out. Let $\mathcal{T}$ be a such family of all sets $U\subset \mathbb{R}^2$ that $U\cap L$ is an open set in L, ...
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86 views

Is the translation of open and closed sets to some language non-antonym preserving?

Maybe more than one person though, before you were given the definition of closed set, that they were the sets that are not open, i.e. that the property of open and closed being antonyms were ...
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128 views

Is this a general structure for constructs?

Here a construct is a category where the objects are sets and the morphisms are structure preserving functions. Common examples are groups, graphs and topological spaces. As far as I can see there is ...
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When is a metrizable topological vector space locally bounded?

Consider a topological vector space $E$ with topology $\sigma$. Suppose that $E$ is metrizable, in other words, that there exists a metric $d$ on $E$ that induces the topology $\sigma$. One can then ...
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An excerpt from a seminar

It is a statement that a professor made in a seminar which I attended yesterday.He says that the following hold: $1$.If $D$ denotes the closed unit disc then there does not exist a continuous ...
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path connectedness of space of almost commuting matrices

Let $R$ be a topological ring which is a domain. Let $n$ be an integer and let $\zeta_n$ be a $n$-th root of unity. Denote by $X$ the set of $m$ by $m$ invertible matrices with coefficients in $R$ ($m$...
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Problem 30 in the Exercises following Chapter 2 in Baby Rudin: How to immitate the proof of Theorem 2.43?

Here's problem 30, the very last one, in the Exercises after Chapter 2 in Walter Rudin's Principles of Mathematical Analysis, 3rd edition: Imitate the proof of Theorem 2.43 to obtain the following ...
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Condition ici=ic on a topological space is equivalent to if each dense set has dense interior in the space.

I am required to prove the following: Let $(X,\tau)$ be a topological space.Then each dense set has dense interior iff $ici=ic$ holds where $i$ is the interior operator and $c$ is the closure operator....
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Proof of Triangulation Theorem for 1-Manifolds

While I am reading "Introduction to Topological Manifolds" by John M. Lee, I come to see the following paragraph in the proof of Theorem 5.10 pp. 102. Note that Int$\ e\cap\ $Int$\ e'$ is open in ...
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Is a countable, nowhere compact, zero-dimensional, dense in itself, Hausdorff space which is 2nd countable; homeomorphic to space of rationals?

Let $X$ be a countable, nowhere compact, zero-dimensional, dense in itself, Hausdorff space which is 2nd countable. Is $X$ homeomorphic to the space of rationals? $X$ is called nowhere compact when ...
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Is $S(\mathbb{R}^{d})$ dense in $L^{1}_{\textrm{loc}}(\mathbb{R}^{d})$?

Let $S(\mathbb{R}^{d})$ denote the class of Schwartz functions in $\mathbb{R}^{d}$. Is it true that $S(\mathbb{R}^{d})$ is dense in $L^{1}_{\textrm{loc}}(\mathbb{R}^{d})$, the locally integrable ...
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28 views

Singleton sets and net criteria for closeness

Theorem. Let $(X,U)$ be a topological space and let $A$ be a subset of $X$. Then $x \in cl(A)$ if and only if there is a net in $A$ that converges to $x$. My question? Does this theorem imply that ...
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Is this statement true?(covering map)

Let $C,X$ be topological spaces. Let $p:C\rightarrow X$ be a continuous function. Let $U$ be an evenly covered open subset of $X$. Let $V$ be an open subset of $C$ such that $p|_V:V\...
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36 views

Showing $\phi(f \cdot g) = \phi(f) + \phi(g)$

For $\phi \in C^1(X; G)$ a cocycle being thought of as a function from paths in X to G, I want to show: $\phi(f \cdot g) = \phi(f) \cdot \phi(g)$. What I'm not sure is how I'm supposed to relate a ...
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Does there exist a continuous function between the following sets:

Does there exist a continuous function between the following sets: $A.f:(-1,1)\rightarrow (-1,1]$ which is onto and one-one $B.f:\{(x,y):y^2=4x\}\rightarrow \mathbb R$ which is one-one What ...