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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37 views

$1 \leq a^2x^2 + b^2y^2 - abxy \leq 9 , x\geq 1$- question compactness and connectedness..

I was told that this object was a cone, I cannot see that, can anyone tell me how to identify which object this is, so as to continue assesing and answering questions of compactness and ...
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57 views

Additional assumptions on function to ensure uniform convergence

Given a sequence $u=(u_n)_{n\geq1}$ converging to $1$, I would like to prove uniform convergence of the sequence of functions $f_n$ defined by $f_n(x)=f(u_n x)$ for $x\in\mathbb{R}_+$ to the function ...
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34 views

commutativity taking the complement and taking fibers

Let $\mathcal M \rightarrow S$ be a projective irreducible scheme over the spectrum of a DVR and $U\subset \mathcal M$ an open subscheme surjective on $S$. Is it true for both points (generic and ...
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30 views

How do I prove that this product space is normal?

Let $A$ be a compact subspace of $\mathbb{R}^2\setminus\{0\}$. Let $C$ be a connected component of $\mathbb{R}^2\setminus A$. Define $D=C\cup A$. How do I prove that $D\times [0,1]$ is normal? I ...
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27 views

Is $\mathbb R^N$ an $C$-distinguished topological space?

I am reading a paper which has some complicated construction on a Hausdorff topological space called $C$-distinguished topological space. The paper says that a $C$-distinguished topological space $X$ ...
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54 views

Cylinder and Möbius strip as fiber bundles: trivializations and cocycles

I know that this question has already been asked, but I couldn't find a clear answer. I have to show that the cylinder and the Möbius strip are fiber bundles over $S^1$ with fiber an open interval ...
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23 views

Specific problem on Radon measures from Folland's real analysis on $ C_0(X) $

Hello all I am trying to understand the concept of $ C_0(X) $ within the concept of Radon measures as presented in Folland's real analysis chapter 7, so far so good right until I came across problem ...
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83 views

Alternative ways to prove $\{f:f(0)=\sum_k f(\frac{k}{\sqrt{n}})g_n (k)\}$ is dense in $\{f\in C^2 (\mathbb{R}) : f(0)=\int_{\mathbb{R}} f(u)g(u)du\}$

I want to prove that $$E:=\bigcap_{n\geq 1} \left\{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f\left(\frac{k}{\sqrt{n}}\right)g_n (k)\right\}$$ is a dense subset of: $$F:=\left\{f\in C^2 (\mathbb{R}) ...
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48 views

Every open set is the union of net of increasing open sets

I'm struggling to find a solution to the following problem: Let $(X,\mathcal{T})$ be an arbitrary topological space and let $\mathcal{U}$ be an class of subsets of $X$, i.e. ...
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35 views

Continuity by composition with a homeomorphism

I only want to know what do you guys think about the following proof. That's an exercise I've tried to do and I don't have an available answer, so... If you find some error or imprecision, I'd be ...
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94 views

Supportive book(s) for unproven-theorems of General Topology by R Engelking?

I am studying General Topology by R Engelking. And, it has many theorems left without proofs. Some of them are very hard and I don't think the author had intention to leave them as exercises. Would ...
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45 views

Subbasis of Product Topology

There's a thing that confuses me about the product-topology. Namely, we said in class that the sets of the form $U_1 \times \cdots \times U_n \times \prod_{i = n+ 1}^\infty X_i$, where $U_i$ is open ...
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51 views

Proof Attempt of Brouwer (via Separating Hyperplane Theorem)

In part motivated by the discussion here, I have been playing with trying to prove Brouwer's theorem appealing as minimally as possible to topology. In the 1-dimensional case I believe one can ...
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51 views

Why (or when) is the direct limit of compact spaces paracompact?

I'm working through Milnor and Stasheff's Characteristic Classes and got stuck in chapter 5, p.66, where some (supposedly) easy facts about paracompact spaces are assembled. One of these is: ...
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22 views

Is the Zariski topology equipped with Eisenstein's metric an analytic submanifold?

Using $M=(C(\mathbb{R}),T_z)$ with the norm $(x,y) \to \log(\partial_x+\partial_y)$, we can easily define a derivative using distributions. I was wondering: Does this make $M$ an analytic manifold ...
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39 views

Prove that set is perfect

Let $E$ be the set of all $x\in [0,1]$ whose decimal expansion contains only the digits $4$ and $7$. Is $E$ perfect? Proof: Here was proved that $E$ is closed set in $[0,1]$ (also in $\mathbb{R}^1$). ...
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64 views

Importance of (the number of) ultrafilters

I have seen a proof about the numbers of ultrafilters in topology, namely a theorem of Pospíšil stating that there are $2^{2^\kappa}$ ultrafilters on every infinite set $X$ with $\kappa = |X|$. The ...
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23 views

Polyhedral surface with infinitely many triangulations with same combinatorics

Is there an example of a polyhedral surface that has infinitely many triangulations with the same combinatorics?
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22 views

Locally compact spaces that are not first-countable and continuity of functions on locally compact groups and continuity of group representation

If $X$ is a topological space that is first-countable, then a function $f: X \to Y$ into another topological space $Y$ is continuous if and only if $f$ is sequential continuous. Only the implication ...
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25 views

Any online videos on a course taught from Munkres?

Are there any vidoes available on the Internet --- for watching online or for download --- of any (general) topology course taught using the book Topology by James R. Munkres, 2nd ed? If so, please ...
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27 views

A basis $B$ for a topological space $X$ is a ring of sets iff $X\in B$

Let $X$ be a topological space with basis $B$, and suppose that $B$ has the following properties: $B$ consists of compact and dually compact$^\ast$ subsets. For every triple $U, V, W\in B$, we have ...
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38 views

Connectedness of circle without center line across it

Using a definition I saw in an old Russian book, a set in $\mathcal R^{n}$ is said to be connected if it cannot be represented as a disjoint union of two nonempty, separated sets. Separated, meaning ...
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83 views

Distance between sets

Let $K \subset K_1 \subset U \subset \Bbb R^2$, such that $K$ and $K_1$ are compact sets, with $K \subset \mathring {K_1}$, and $U = \mathring U \subsetneq \Bbb R ^2$. If $w \in \partial K_1$ such ...
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24 views

What is the formalism for a map that returns the adjacent vertex positions of a given adjacency matrix?

How do I formally denote a map that returns the adjacent vertex positions of a given adjacency matrix? Example: V = undirected adjacency matrix $$ V = \left[ \begin{matrix} v_{1,1} & v_{1,2} ...
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19 views

Proposed proof for quasi-metric result

A quasi-metric on a set $X$ is mapping $\rho: X \times X \rightarrow [0, \infty)$ satisfying the following conditions: $\rho(x,y) \geq 0~~\text{and}~~\rho(x,x) = 0;$ $\rho(x,z) \leq \rho(x,y) + ...
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41 views

Proof of Supporting Hyperplane Theorem from basic definitions.

My purposes in posting this question are twofold. First, I would like to have a lemma which I have proven on the way to proving the Supporting Hyperplane Theorem checked for rigor (zero tolerance for ...
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22 views

Construct an example in which $x$ is $\tau_1$-accumulation point of a subset $A$ of $X$ but It is not $\tau_2$-accumulation point of $A$

Let $\tau_1$ and $\tau_2$ be a topologies on a set $X$ with $\tau_1 \subset \tau_2$ Construct an example in which $x$ is $\tau_1$-accumulation point of a subset $A$ of $X$ but It is not ...
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32 views

Weakening compactness in metric spaces

Is the following true in a general metric space $X$? Every net (in $X$) of cardinality $\kappa$ contains a convergent subnet of cardinality $\kappa$ if and only if every open cover of $X$ admits a ...
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38 views

If $x \in A$ is a limit point of $A$, does there always exist an $\epsilon$-nhood of $x$ for every $a \in A$ such that $a \in V_{\epsilon}(x)$?

The definition of limit point that I'm using is: Definition. A point $x$ is a limit point of a set $A$ if every $\epsilon$-neighborhood $V_{\epsilon}(x)$ of $x$ intersects the set $A$ in some point ...
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53 views

General topology by Bourbaki

I have a very vague question , I have taken a first course in general topology (first four chapters of Munkres's Topology), now I want to learn more general topology. I heard Topology by Bourbaki is ...
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43 views

Inclusion in cone is homotopy equivalence

Suppose $X$ is a topological space and $x \in X$. Let $CX$ be the cone of $X$, i.e. the quotient space $X \times [0,1]/{\sim}$ where $(x,1) \sim (y,1)$ for alle $x,y \in X$. I would like to show that ...
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19 views

How to show explicitely that 2-sheeted covers are Galois?

Let $X,Y$ be connected Hausdorff topological spaces. It is well-known that every 2-sheeted covering $p:Y\to X$ is Galois which means that $Aut(Y/X)$ acts transitively on fibers. It is easy to come up ...
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15 views

Quickest way to restrict a homeomorphism

Let $\phi: U \to V \subset \mathbb{R}^n$ homeomorphism. My desire is: I want to say the restriction $\phi|_{\phi^{-1}(B_{r'}(x))}:\phi^{-1}(B_{r'}(x)) \to B_{r'}(x) $ is a homeomorphism in the ...
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46 views

$f$ is continuous $\iff f(\bar A) \subset \overline{f(A)}$

The problem is: $f:X\to Y$: any map. $f$ is continuous $\iff \forall A\subset X, \ f(\bar A) \subset \overline{f(A)}$ My understanding is: Suppose $f$ is continuous. $\forall A\subset X, A ...
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23 views

Boundary of surface

Let $S$ be the region in $\mathbb{R}^2$ bounded by $x$-axis, $x=1$, and $y=x$. Define $$ f(x,y) = \begin{cases} 0 &\mbox{if } x = 0 \text{ or if $x$ or $y$ is irrational} \\ 1/q & \mbox{if ...
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35 views

Help in Understanding the Proof of Baire-Category theorem

In the proof of the Baire category theorem(for non-empty Banach Spaces), I cannot understand the following Baire Category Theorem: A non-empty Banach Space cannot be a countable union of nowhere ...
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41 views

closedness of compact sets in some topological spaces

Is there any famous axiom on X other than Hausdorffness or axioms leading to Hausdorffness,such that every compact set in X is closed?
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34 views

Question about limit points

The reference I'm using defined a limit point of a set as: Definition. A point $x$ is a limit point of a set $A$ if every $\epsilon$-neighborhood $V_{\epsilon}(x)$ of $x$ intersects the set $A$ in ...
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39 views

Existence of an homeomorphism between $X$ a complete separable metric space and a subspace of $[0,1]^{\mathbb{N}}$

Result: If $X$ is a complete separable metric space then there is a $E \subset [0,1]^{\mathbb{N}}$ such that $X$ is homeomorphic to $E$ ($E$ is a $G_\delta$ set - is the intersection of denumerable ...
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47 views

Differences between a quotient map and a continuous function in topology

Def. for a continuous function: Let $X$ and $Y$ be topological spaces. A function $f : X \rightarrow Y$ is continuous if $f^{-1} (Y)$ is open in $X$ for every open set $V$ in $Y$. Def. for ...
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44 views

Intersection of an open and closed set

Suppose we have $A\cap B=C$ where $A$ is closed and $B$ and $C$ are open. Does this imply anything else about the sets or their limit points? My intuition says it might imply something like $A\subset ...
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31 views

Question about Proof of 3.15 in Rudin's Functional Analysis

In Rudin's Functional Analysis, the following claim is made (Banach-Alaoglu theorem): If $X$ is a TVS and $V$ is a nbhd of $0_X$, then $$ K_V \equiv \{ \Lambda\in X^\ast | |\Lambda(x)|\leq1\forall ...
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17 views

Is the unitary group of a pre Hilbert space contractible?

for a separable Hilbert space $H$ it is known that the unitary group $U(H)$ is contractible, both for the norm topology (Kuiper's theorem) and for the strong operator topology (Dixmier and Douady, ...
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25 views

What about paracompact operators between Banach and quasi-Banach?

Reading a previous post I ask to me, by curiosity, about a reasonable notion of paracompact operator. Paracompactness in mathematics is a property that have good performances. Let $T:X\rightarrow Y$ ...
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16 views

Finitely many hyperplanes separating $ x,y $ in a CAT(0) cube complex

I'm having a great difficulty understanding a proof of a lemma from this paper: http://www.math.hawaii.edu/~erik/papers/cat0-A.pdf It's lemma 1.12. To make it shorter for anyone who'd like to take a ...
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31 views

Given a set of points, describe the structure(number of holes, etc.)

If we stay in two dimentions, we may consider graphs. Suppose we have a set of points abc bcd ak kd (a generic example) How may one go about counting the holes in the structure, and it's general ...
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48 views

Proof of the Arzela-Ascoli theorem - where is the assumption that $X$ is compactly generated used?

I'm learning the proof of the following version of Arzela-Ascoli's theorem (Willard, General Topology, page 287): Let $X$ be a Hausdorff, or regular, k-space, $(Y,\mathcal D)$ a Hausdorff uniform ...
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101 views

Why do we need tube lemma to prove the compactness of the product of two compact spaces?

I read the proof in Munkres' book Topology which uses the tube lemma but still thinking about an easier proof using basis of product topology : $X \times Y$ has $$\{B_x \times B_y, B_x \times Y, X ...
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38 views

What is $\Bbb E$? Is is $\Bbb R$ with the standard Euclidean topology?

What is $\Bbb E$? I believe this is just alternative notation for $\Bbb R$ where $\Bbb R$ is assumed to have the standard Euclidean topology. Is this correct? Is seems to be used in this way in ...
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47 views

$SO(3)$ vs 3-Torus

$SO(3)$ and 3-Torus both can be viewed via rotations for a rigid body. They are not diffeomorphic. $SO(3)$ can be decomposed into three axial rotations. Could I think the reason they are not ...