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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A confusion of a real analysis online lecture: Relative compactness

https://www.youtube.com/watch?v=kkKfRaI-cqs At 13.00 what does the professor mean to let those subcovers be "restricted" to Y? Is that a process like A is contained by B implies that A intersect C in ...
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22 views

Confused about topological vector space defined by seminorms

I'm reading a book in which it's claimed that for a strongly continuous representation, $U: G\rightarrow Aut(E)$ of a Lie group, G, on a locally convex, complete, Hausdorff topological vector space,E, ...
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21 views

Looking for a paper on Weakly uniform bases

I want to find an old paper: R.W. Heath, R.W. Lindgren, Weakly uniform bases, HOUSTON JOURNAL OF MATHEMATICS. 2(1) (1976) 85–90 Could someone help me? A link is also welcome. Thanks!
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37 views

Boundary disconnects connected space

I need to show that a boundary of an open subset $\emptyset \neq A\subset X$ whis isn't dense in a connected space X, disconnects X. Any hints how should I start?
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37 views

$f:X \to F(Y)$ is measurable iff $f^{-1}(\{\emptyset\})$ is measurable and there is a sequence of measurable functions

Let $X$ be a measurable space and $Y$ a Polish space. I want to show the following equivalence. $f:X \to F(Y)$ is measurable iff $f^{-1}(\{\emptyset\})$ is measurable and there is a sequence of ...
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33 views

contractivity to prove that an entire function is of finite exponential type?

Suppose that $G(z)$ is an entire function. If I can show that G is contractive within some compact and convex subset of $\mathbb C$, is it easier from that point to establish that $G$ is of finite ...
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21 views

Necessity of continuity in Topological Vector Space

In the notion of a topological vector space, we define such as a vector space $X$ (over a field $\mathbb{K}$) with topology $\mathscr{T}$ such that $$\iota_+: (X,\mathscr{T}) \times (X, \mathscr{T}) ...
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46 views

How to prove that the topological spaces are homeomorphic

Let's consider a topological space $X$. We define $X \times I / {\sim}$ as a product of $X$ and a $I=[0, 1]$ closed interval's quotient by the following equivalence relation: $(x, 1) \sim (y, 1), (x, ...
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25 views

Is either of these subsets connected in $\mathbb{R}^\omega$ in the product, uniform, box topologies?

Let $\mathbb{R}^\omega$ be the set of all (infinite) sequences of real numbers. Let $A$ be the set of all bounded sequences of real numbers, and let $B$ be the set of all unbounded sequences. Then ...
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31 views

Closed mapping theorem for almost connected groups

Edit: I originally asked if $\phi$ is a closed map. Clearly it isn't in general, as the real numbers can be mapped to the torus with dense image. Let $G$ and $H$ be connected locally compact groups, ...
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62 views

Topological Conjugacy between tent and skewed tent map

Consider the family of skew tent maps $\mathcal{S}$ on $[0,1]$, such that: $S(0)=S(1)=0$; The peak (maximum) of the tent occurs at $S(a)=b$; $\max(a,1-a)<1$ which implies the map to be locally ...
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36 views

Proof of Generalized Stokes Theorem Stuck

I've been slowly working my way through understanding the proof for the Generalized Stokes' Theorem but I've hit a snag from one of the Propositions. In the attached image, I've been able to ...
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39 views

Prob. 12, Sec. 23 in Munkres' TOPOLOGY, 2nd ed: How are these unions connected?

Let $Y \subset X$; let $X$ and $Y$ be connected. Suppose that $A$ and $B$ form a separation of $X - Y$. Then how to show that $Y \cup A$ and $Y \cup B$ are connected? My effort: Now since $A$ ...
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42 views

Function not equal a.e. to continous function on real line and on circle

I am looking for a proof of the following fact: Suppose that $H: \mathbb{R} \rightarrow \mathbb{R}$ is a periodic function with period $1$. Suppose further that there is no continuous function ...
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26 views

Proving that local base determines topology.

Let $(X,\tau)$ be a topological vector space and $\mathcal{B}$ a collection of neighborhoods of $0$ such that every neighborhood of $0$ contains a member of $\mathcal{B}$ (that is, $\mathcal{B}$ is a ...
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17 views

A non-zero linear functional can induce a positive linear functional?

A non-zero linear functional can induce a non-trival positive linear functional? $X$ is a topological sepace, if $ f:C_{c}(X)--->R $, is a linear functional, and $f$ is not always equal to zero, ...
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22 views

About a map between two topological manifolds with different dimensions

Let $M_1$ be a $n$-dimensional topological manifold and let $M_2$ be a $m$-dimensional topological manifold, such that $m>n$. Moreover, let $U\subset M_1$ be an open set and let $f:U\rightarrow ...
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28 views

Operator norm on the space on linear functions between Euclidean spaces.

*I'm reading a text which has a preliminary section on Linear maps. I have come across a conclusion that I can't seem prove by myself. * Let $Lin(\mathbb{R}^m,\mathbb{R}^n)$ be the space of linear ...
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54 views

Classifying space infinite totally ordered set contractible

Let $(X,\le)$ be a totally ordered set. Regarding it as a category, it has a classifying space $B(X,\le)=|N_\bullet(X,\le)|$. This should be the (possible infinite) simplex with vertices $X$, hence I ...
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42 views

$\{P\}^-$ notation in Algebraic Geometry

I am reading proposition 2.1.6 in Algebraic Geometry by Robin Hartshorne. Near the end of the paragraph, he defines a function $ \alpha:X \rightarrow t(X) $ where $X$ is a topological space and $t(X)$ ...
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27 views

Interpretation of a weakly compact set of functions

I'm having trouble really grasping the idea of a weakly compact set. The set I have under consideration is a set of functions $M_c$, where $$M_c=\{f:W(f)\leq c\},$$ where $c\geq0$ and $W(f)$ is a ...
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35 views

Every point in a Tychonoff Space is contained in a compact set

This should be very elementary, but I just can't see it: Is every point in a Tychonoff Space contained in a compact set? I tried to look for a counterexample and figured that it cannot be locally ...
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29 views

Prob. 4, Sec. 21 in Munkres' TOPOLOGY, 2nd ed: How to decide which cases to consider?

We need to show that the ordered square satisfies the first countability axiom. I'm not able to decide as to which separate cases to consider. By definition the ordered square is the product $I ...
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43 views

Solution Verification: Prove in detail that the open rectangles in the Euclidean plane form an open base

I want some verification and/or some polishing on my proof. However if it is good, please let me know (I think this is highly unlikely to happen). Problem. Prove in detail that the open rectangles ...
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23 views

Homomorphisms between countable spaces and Euclidean spaces?

Is there some place to start reading about homomorphisms between countable (discrete) spaces and Euclidean spaces or $l_2$? I know it is a rather general question, but I am not sure what I am looking ...
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26 views

Using Bounded Operator sequence Theorem

Let E$\subset L_1$ be a set of fourier series functions $e_n(t)=e^{int}$ for $n \in Z$. What is meant by saying to prove $Ge_n$ is a scalar multiple of $e_n$ and it is continuous? How can we prove it? ...
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13 views

Non-rectifiable bounded closed set

Find a bounded closed set in $\mathbb{R}$ that is not rectifiable. When I say "$S\subset \mathbb{R}^n$ is rectifiable" I mean that the constant function $1$ is Riemann-integrable over $S$ (then one ...
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43 views

Closure, Interior, Boundary and Exterior of a Set in different topologies…

Closure, Interior, Boundary and Exterior of a Set in different topologies... This is something I seem to be majorly struggling with I am looking at the set of all $A = \mathbb R - \mathbb Q $. I ...
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26 views

Prove every bounded sequence in the real numbers has a convergent subsequence using cauchy

Prove that every bounded sequence in $\mathbb{R}$ has a convergent subsequence. I am going to use the Cauchy sequence property to do this. Proof: Let $A_n$ be a bounded sequence in $\mathbb{R}$. ...
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22 views

Proving that if the real numbers are complete than every cauchy sequence converges

Prove that if $\mathbb{R}$ is complete then every Cauchy Sequence converges. Definition of complete: A set $A$ of real numbers is said to be complete if every Cauchy sequence $<a_n\in ...
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33 views

A problem about general topology. (homeomorphims)

Let $f:X \to Y$ homeomorphims and $A \subseteq X$ have the property that $A \cap A^{'}=\emptyset$. Then $f(A)\cap (f(A))^{'}=\emptyset$. Idea: Contrapositive. Suposse $f(A)\cap (f(A))^{'} \neq ...
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33 views

Proving that $S^n$ (n-sphere) is locally connected.

Definition: A space X is said to be locally connected at $x\in X$ if for any open set $U$ containing $x$, there is an open connected subset of $U$ (say $W$) containing $x$.$$x\in W\subseteq U$$ A ...
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29 views

Prove $X \setminus \operatorname{Cl}A = \operatorname{Int}(X \setminus A)$

Definitions ($X$ is a topological space): • $\operatorname{Cl}A$ is the intersection of all closed subsets of $X$ which contain $A$ as a subset. • $\operatorname{Int}A$ is the set of all ...
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33 views

Weak topology and the closed unit ball

I want to prove that there is no neighbourhood of $0$ in the closed unit ball. I can use pointwise and Banach-Alaoglu theorems to prove it.
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42 views

Prove that this infinite sum involving metrics is also a metric

The following is a question from a previous assignment that I was unable to complete. Any assistance on how to complete this would be appreciated. Let $\varrho_i: X\times X\to \Bbb R^+$ with ...
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56 views

Hausdorff, regular and separable space

Let be $X$ a topological space such that X is a Hausdorff, regular and separable space. If $U\subseteq X$ is open such that $U=int(cl(U))$, and $E\subseteq X$ is a countable dense set, I need to prove ...
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33 views

Precompact and locally finite implies finite intersection

An exercise in Lee's Introduction to Smooth Manifolds asks the following: Let $M$ be a topological manifold, and let $\mathcal U$ be an open cover. Suppose the sets in $\mathcal U$ are precompact ...
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28 views

Good partition in locally compact metric space

Let $X$ be a locally compact second countable metric space, endowed with a Borel probability space $\mu$. Let $\varepsilon>0$. Question: Is it possible to find a countable Borel partition $\xi$ of ...
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25 views

Identification of polygon edges

In Klein's famous example of regular 14-gon made of 336 copies of (2,3,7) triangles, he used identification for edges such that side 2i+1 is identified with side 2i+6 (mod 14). But I wonder how could ...
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18 views

Weak-* closure and convexity

I'm trying to write a proof of Goldstine's theorem : the weak-* closure of the unit ball of a normed vector space $X$ is the unit ball of the second dual $X^{**}$. At some point I would like to use ...
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43 views

Concept of Boundedness

I noticed there are two notions of boundedness, one in the context of order theory and other in the context of metric spaces. In a metric space (X,d) , we talk about subsets of X being bounded iff ...
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34 views

Show that the function sending $f$ to its derivative is continuos in compact-open topology.

Let $\Omega$ $\subset$ $\mathbb{C}$ be an open subset of the complex plane. We need to show that the function $D$: $\mathcal{H}(\Omega)$$\rightarrow$$\mathcal{H}(\Omega)$ $f \rightarrow f'$ is ...
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46 views

Show that there is a Borel function from [0,1] to $2^ω$

Show that there is a Borel function from [0,1] to $2^ω$. Conclude that there is a Borel function from $[0,1]^ω$ to $2^ω$. I'm not sure how to go about this. I know that a function is Borel if the ...
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23 views

Find the accumulation points of $A = \{ (x,y) : y=0 \}$ in $\mathbb{R}^2$.

Find the accumulation points $A^\prime$ of $A = \{ (x,y) : y=0 \}$ in $\mathbb{R}^2$. My attempt: Take any $p \in \mathbb{R}$ $\implies (p,0) \in A$. Now we can see $\forall r > 0, B^\prime ...
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61 views

Proof that an uncountable set with the countable complement topology is not first countable

In an exercise set for my topology course, I was asked to prove that an uncountable set with the countable complement topology is first countable. I couldn't prove it, and I started to suspect that it ...
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29 views

Question about diameters

$\newcommand{\diam}{\operatorname{diam}}$If we take a compact set $K$ in $\mathbb{R}^n$ and $\varepsilon>0$ how we can prove that $$\diam(K+\varepsilon B^n)=\diam(K)+\varepsilon\diam(B^n)\text{ ...
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26 views

Is this a homotopy between the path $\alpha^{-1} \cdot \alpha$ and the constant path?

Given some path $\alpha(s)$ from $x$ to $y$ in a topological space, I want to show that the product of the path and its inverse, $\alpha^{-1} \cdot \alpha$, is homotopic to the constant path $e(s)$ ...
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18 views

Finitely generated modules over completed group rings

Let G be a profinite group. Let $M$ be a finitely generated $\Bbb{Z}_p[[G]]$ (completed group ring). How to show the following facts: $M$ carries a unique Hausdorff topology(called its canonical ...
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24 views

Product $\sigma$-algebra on $\mathbb R^{\mathbb N}$

Let $\mathbb R^{\mathbb N}=\mathbb R\times\mathbb R\times\ldots$ be the space of all real sequences and endow it with product topology. Is the product $\sigma$-algebra generated by Borel subsets of ...
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21 views

Description of a universal regular space

tl;dr: Given a topological space $X$, is there a useful description of the universal continuous map from $X$ to a regular (Hausdorff) space? The Čech–Stone compactification $\beta X$, given a ...