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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Irrational winding of a torus

Let $f: \mathbb{R} \mapsto T=\mathbb{R}^2/\mathbb{Z}^2$ such that $x \mapsto (x,\alpha x) \, \, \mod \mathbb{Z}^2$ and $\alpha$ irrational. I want to prove that $f$ isn't an embedding. In order to do ...
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36 views

Tikhonov theorem and $L^1$ completeness

The idea is to prove completeness of $L^1$ using Tikhonov theorem. The proof will be for narrower class of functions though. There is a Tikhonov theorem that states that for any set of compact ...
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32 views

Open cover of manifold with boundary

If $\mathcal{O}=(U_{\alpha})_{\alpha\in A}$ is an open cover for a smooth manifold $M$, then each $U_{\alpha}$ is a smooth manifold. I want to extend this fact to manifolds without boundary. So my ...
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99 views

Connectedness of the Complement of the Countable Union of Closed Jordan Regions

Let $\{V_j\}$ be a countable collection of pairwise disjoint closed Jordan regions in $\mathbb{R}^2$. That is closed sets whose boundary is a Jordan curve. Let $$U = \mathbb{R}^2 \setminus \bigcup ...
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46 views

Perfect preimages of compact spaces are compact

Below is a question concerning compact spaces from James Munkres' Topology. Following that is my attempt at a solution, which I am not sure is correct and would appreciate if somebody could point out ...
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31 views

Isomorphisms with invariant linearly independent dense subset.

If $T$ is an isomorphism acting on a separable Banach space $X$, can we find a countable, dense, linearly independent set $D\subset X$ such that $T(D)=D$? If $X$ is finite dimensional, then the answer ...
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49 views

about Heine-Borel Theorem in a function space

In Pugh's real mathematical analysis. About the Heine-Borel Theorem in a function space, it states that a subset $\epsilon$ $\in C^0$ is compact if and only if it is closed, bounded, and ...
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33 views

Can someone intuitively describe the fiber bundle and product spaces of SO(3)?

I have zero understanding of differential geometry or topology so the material found online are useless for me. So in light of that can someone use very general terms or analogy to comment about the ...
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30 views

homotopy between continuous functions to an absolute retract

I have the following statement to prove as one of the "fundamental" questions our topology professor wants us to know for his final: Let $X$ be a topological space, and let $A$ be an absolute ...
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10 views

Parametrizing regions of complex plane

Let $\Omega=\mathbb{C}\setminus \lbrace t e^{it} \ \vert t \in \mathbb{R}_{\geq0} \rbrace$ I need to write $\Omega= \coprod_{i=0}^{\infty} R_i$ where each $R_i$ is the region bounded by from $t=2k ...
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22 views

Two non-homeomorphic spaces with continuos bijective functions in both directions

I was asked the following question: if two topological spaces $X, Y$ are such that there exist a function $f:X\rightarrow Y$ continuos and bijective and a function $g:Y\rightarrow X$ continuous and ...
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32 views

Density character of a metric subspaces

Is it true that if $M$ is a metric space and $N$ is a metric subspace of $M$ (I mean, $N\subseteq M$ and the metric defined on $N$ is the same metric on $M$ restricted to $N$) then the density ...
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29 views

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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70 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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27 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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37 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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33 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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62 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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76 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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51 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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39 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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40 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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40 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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33 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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49 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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18 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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42 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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30 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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69 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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25 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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32 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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62 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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36 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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48 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 ...
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43 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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34 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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55 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 ...