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 over $C^0(\mathbb{R})$

Let $C^0(\mathbb{R})$ be the set of continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$, For any continuous function $h > 0$ consider $B_f(h) = \{ g \in C^0(\mathbb{R}) : |f(x) - g(x) ...
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68 views

Equivalence of sigma algebras on the set of probability measures.

I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and ...
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33 views

how to conclude a subset of $M_n(\mathbb{C})$ is compact from spectral radius?

could any one tell me which of the following is/are compact subset? $S=\{A\in M_n(\mathbb{C}): \rho (A)\le 1\}$ $S=\{A\in M_n(\mathbb{C}): A=A^*,\rho (A)\le 1\}$ $S=\{A\in M_n(\mathbb{C}): ...
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23 views

Entropy of isometric extension

A similar question to mine was asked before at the address below but it was not answered there so I am asking it again. Also there is a more specific case I am interested in. Topological entropy of ...
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35 views

Topological spaces from compact Hausdorff zero dimensional spaces

I saw a construction of general topological spaces using compact Hausdorff zero dimensional topological spaces, but I have no clue now of the construction or reference to this. I would be thankful if ...
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31 views

Test functions are dense in $L^p$?

I was wondering about the following: If we say that the test functions are dense in $L^p$, does this imply that there is also always a sequence of them converging pointwise and in $L^p$ norm to such a ...
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56 views

Separable spaces and functions that separate points

In a metric space, does existence of a function that separates points imply that the space is separable and conversely? I'm just a baby Rudin student. Thanks in advance for every hint.
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75 views

Prove that if $T$ is one-to-one on $D$, then the set $T(D)$ is open

Let $f$ and $g$ have continuous first-order partial derivatives on an open set $ D\subseteq\mathbf{R}^2 $ and let $T :D \to \mathbf{R}^2 $ be defined by $ T(u,v)=(f(u,v),g(u,v)). $ ...
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34 views

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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31 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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25 views

simplicial approximation and infinite complexes

It is well known that if $X$ is a finite simplicial complex then for every continuous map $f:|X|\to |Y|$ there exists a simplicial map $F: X^{(n)}\to Y$ that $|F|$ is homotopic to $f$. Does anyone ...
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49 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 $$ ...
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36 views

Weakly closed $\iff$ closed using the Separation theorem

My question is about the following problem. $X$: Banach space, $C$: convex subsets of $X$. Then, followings are equivalent. i) $C$ is closed. ii) $C$ is weakly closed. I ...
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31 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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39 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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38 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 ...
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28 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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44 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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33 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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30 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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27 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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46 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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16 views

Compactification: density of a uniform space $X$ in the spectrum of $UC^b(X)$

First, a small motivation: Suppose we are looking for a compactification of uniform spaces, satisfying an universal property similar to the one of the Stone-Čech compactification of a locally compact ...
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40 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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29 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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50 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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65 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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23 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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75 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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27 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 ...
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52 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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61 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 ...
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34 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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$E$ 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 ...
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39 views

Showing that an evaluation map is continuous

This is a problem from Munkres' Topology 43.8 If $X$ and $Y$ are spaces, define e : $X \times \mathscr {C}(X,Y) \to Y$ by the equation e($x,f$) $= f(x)$; the map e is called the evaluation map. ...
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33 views

No norm consistent with given topology

Given the (Frechet) topology on the Schwartz class $S(\mathbb{R}^d)$ induced by the seminorms $\rho_{\alpha \beta}f = \operatorname{sup}_{x \in \mathbb{R}^d}|x^{\alpha}\partial^{\beta}f|$, how can I ...
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54 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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59 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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47 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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31 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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37 views

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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65 views

Isomorphism for the group of units of the ring of integers of a local field

Let $K$ be a local field with a discrete and non-archimedean absolute value, $\mathcal{O}_K$ be its ring of integers, $\mathfrak{m}_K$ be the unique maximal ideal of $\mathcal{O}_K$ and ...
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53 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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28 views

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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45 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)] = ...
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use picture to show the conjugate of a loop in the fundamental group of a space

Give an example of a space in which a loop has a conjugate that is not equivalent to the original loop. this is another question but we did not need work on this
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58 views

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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79 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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39 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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56 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, ...