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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116 views

Twisted tori: discrete and continuous

Taking the advice of Mariano Suárez-Alvarez, I moved this question from MO to MSE: Motivation Let me introduce twisted (discrete) tori: Consider the Cartesian graph product $\mathcal{C}_n = C_n ...
2
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62 views

Continuity of linear form

Let $E=\mathbb{R}[X]$ We define $N:\, P \to \sum_{n=0}^{\infty} { |P^{(n)}(n)|}$ ($P^{(n)}$ being the $n$-th derivative) , it is not hard to prove that $N$ is a norm on $E$. Help me to study the ...
2
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193 views

Separable Banach Space

Let $X$ be a real separable Banach space. Let $A\subset X$ be the enumerable set, given by the separability. How can i define a continuous "bijective" function $f:A\rightarrow\mathbb{Q}$, where im ...
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62 views

Show that a node of a real curve is homeomorphic to a cross

This is related to this question. Let $C = \{(x,y) \in \mathbb{R};\; x^3 + x^2 - y^2 = 0\}$ equipped with the subspace topology of the euclidian plane. I want to show that there's a neigbourhood $U$ ...
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100 views

Separating points from open sets in a compact space without isolated points

Given that $S$ is compact and it has no isolated point. Show that given any nonempty open set $P$ of $S$ and any point $x\in S$, there exists a nonempty open set $V\subset P $ such that $x\notin \bar ...
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112 views

Form of weakly continuous linear functional

This was originally a problem in Stratila and Zsido's "Lectures on von Neumann algebras" (E.1.2). I've spent so much time working on it, and right now I cannot see how the result can be so simple. ...
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40 views

Initial topology of the spectrum mapping $\sigma$

Let $\mathcal{A}$ be a Banach algebra, the map $\sigma$ maps each element $a\in\mathcal{A}$ to its spectrum $\sigma(a)$, which is a compact subset of $\mathbb{C}$. The collection of compact subsets ...
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35 views

Determining whether a path must pass through a given set

I have sets $E^1, \dots, E^n \subset \mathbb{R}^n_{\ge 0}$, none of which contain $0$. I would like to determine whether or not these sets form an "overhang" of $0$. Intuitively speaking, this is an ...
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117 views

Ways to decompose a torus for finite element method so that each cell contains a complete revolution of the major radius

I've got a finite element problem involving paths around the interior of a torus. For this particular problem I think I could make things more computationally efficient if each cell in the mesh made ...
2
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180 views

Fiber Bundle: Hairbrush

I am trying to understand the hairbrush example of a fiber bundle from the Wikipedia article on fiber bundles. If I am understanding this, in the hairbrush example E is the hairbrush, ie. all the ...
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4k views

Rudin 2.2: Prove the set of algebraic numbers is countable.

Similar to Proving that the set of algebraic numbers is countable without AC "A complex number $z$ is said to be algebraic if there are integers $a_0,\dots,a_n$, not all zero, such that $$ a_0 z^n + ...
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66 views

Pointfree generalization of uniform spaces?

Topological spaces generalize as frames and locales. But are there a pointfree generalization of uniform spaces?
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132 views

Euler characteristic of structure sheaf of symmetric product

I recently asked about calculating the Euler characteristic of the symmetric square of a space. There we determined that for a sufficiently well-behaved space $X$ there is a formula $$\chi(X \times ...
2
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173 views

Connectedness of the complement of a compact “small” subset of $\mathbb R^n$

Let $C$ be a compact subset of $\mathbb R^n$ and suppose that for every $\varepsilon >0$ there exists a finite family of open disks $B_i$ s.t. $C \subset \bigcup_{i} B_i$ and $\sum_i r_i \le ...
2
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132 views

When is the projective limit of a sequence of sequential LCVS sequential?

A topological space $X$ is sequential if it is true that an any map whose domain is $X$ is continuous iff it is sequentially continuous. For a projective countable spectrum of sequential topological ...
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453 views

Subharmonic/Superharmonic Inequality in Gilbarg/Trudinger [Section 2.8]

This is in Section 2.8 of Gilbarg and Trudinger. I believe there are some inaccuracies in the proof supplied, and in any case I think there is a more straightforward proof. Definition: A ...
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51 views

Topology of $(\mathcal{A},*)$ determined by $\mathcal{A}_{sa}$?

Let $(\mathcal{A},*)$ be a $*$-algebra, we have the following observation: Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two norms on $\mathcal{A}$ such that the involution is an isometry with respect to ...
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393 views

Recovering the topology of an affine scheme from the specialization preorder

Let $A$ be a commutative ring. The specialization preorder on $\mathrm{Spec}(R)$ is given by $\mathfrak{p} \prec \mathfrak{q} \Leftrightarrow \mathfrak{p} \in \overline{\{\mathfrak{q}\}} ...
2
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68 views

Unit partition to produce smooth function from continuous ones

Given a positive continuous function (except on closed set, where is zero ) on a smooth manifold how to find a smooth function under the same conditions being less (or equal) than this one ...
2
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0answers
73 views

$\sigma$-product space and $ \Sigma$-product space

Recently, I'm interested in the $\sigma$-product space and $\Sigma$-product space. Is there a survey on $\sigma$-product space and $\Sigma$-product space, which is simple for a beginner? Thanks for ...
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143 views

Smooth deformation retracts

Under what circumstances can it be concluded that if two items from the smooth category are related by a topological relationship, then they are also smoothly related in the corresponding way? For ...
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182 views

Countable Product of discrete spaces

Let $X$ be a countable discrete topological space. Consider $X^{\mathbb{N}}$ endowed with the product topology. How do you prove that $X^{\mathbb{N}}$ is homeomorphic to the sub-space of all ...
2
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231 views

Can we find a nice definition of Congruence in Topology?

According to my knowledge, quotient structure is a original structure divided by a congruence. However, quotient topology space is defined this way. Quotient_topology In this way, $\sim$ is only said ...
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108 views

Berkovich analytification of Robinson fields

Let $\rho$ be an infinitesimal and let $^\rho \mathbb{R}$ be a (non-archimedean) Robinson valued field. Is there anything known about the topological structure of $\mathbb{A}^{1,an}_{^\rho ...
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842 views

Are the continuous functions pointwise dense in the bounded measurable functions

Suppose we have a compact set $K$. I know that the space $C(K,\mathbb{C})$ of continuous functions is complete with respect to the norm $\|f\| = \sup_{x\in K} |f(x)|$. Let $L^{\infty}$ be the space of ...
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85 views

JSJ-decompositions of groups and 3-manifolds: a reference request

I am, for whatever reason, interested in learning about the JSJ-decomposition of groups. Having asked around a bit, it was suggested I first learn about what is happening in the manifolds and then ...
2
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0answers
549 views

Showing that metric induces single unique topology on a finite set

I am trying to prove, that given a metric on a finite set it induces exactly one topology. I have an idea which might lead to a proof, but am not sure: For a finite set X with a given metric d we can ...
2
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89 views

Total sets in $R$ compared with total sets in $[a,b]$

A total set in a NLS is one whose linear span is dense in the set. e.g. $A = \{1,x, x^2,...\}$ is total in $(C[a,b],\Vert\cdot\Vert_{\infty})$ I find it easier to talk about total sets than dense ...
2
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110 views

Image of Thom Class under Sequence of Maps?

So I've been trying to do problems in Milnor & Stasheff's Characteristic Classes as a quick review, not having done anything with them in a while. However, I'm stuck on some parts in attempting ...
2
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389 views

The topology of distributions

I have been wondering about the following concerning the spaces $\mathcal D$ of test functions (say on $\mathbf R$). It is my understanding that the topology on this space is inductive limit ...
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565 views

Topology of wedge products

I have a question about the quotient topology induced on the wedge sum $S^{\,2} \vee S^1$, (where $S^n$ denotes the unit sphere in $\mathbb{R}^n$). In this topological space, the subsets $S^1$ and ...
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109 views

A Heegaard splitting of $S^2\times S^1 \# S^2\times S^1$.

For a Heegaard splitting of $S^2 \times S^1$, we can take two copies of genus 1 handlebodies and glue boundaries with the identity map. I want to generalize this a little bit. In the case of ...
2
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268 views

Relations and differences between outer/inner limit and Kuratowski limsup/liminf

Let $X$ be a topological space. I am asking about the relations and differences between the following two different types of $\limsup$ and $\liminf$ of $A_n ⊆ X, n ∈ \mathbb{N}$, a sequence of ...
2
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353 views

Vector calculus- vector field and path

Let $U\subset \mathbb{R}^2$ be open, and $F:U\to \mathbb{R}^2$ a $C^1$-vector field. Assume that: $$\frac{\partial{F_1}}{\partial{x_2}}(x)=\frac{\partial{F_2}}{\partial{x_1}}(x)\quad\forall x\in U$$ ...
2
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92 views

factor of covering map is a covering map?

A paper I'm trying to understand uses the following lemma: Let $p: U \to U_0$ be a topological covering map. Suppose that we can write $p =\pi \circ f$, where $f:U \to Y$ is an open surjective map, ...
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94 views

topology exercise. compactness circle projective space.

Is the circle compact in $\mathbb{P}_{2}(\mathbb{C})$? Here what I did: I considered the circle in $\mathbb{C}^2$ is $\{(x,y)\in\mathbb{C}^2|x^2+y^2=1\}$. The projective closure in ...
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165 views

Trying to prove a intuitively “obvious” fact.

I'm trying to prove that all continuous maps of pairs $f:([-1,1], \{-1,1\})\to (\{-1,1\},\{-1,1\})$ are constant, and I've almost got a working argument, but it reduces down to the following ...
2
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0answers
81 views

Does it imply a lift?

Let $p:S^1\times S^3\rightarrow S^1\times S^3$ be a covering map with $p(z,y)=(z^3,y)$ and $z\in S^1\subset\mathbb{C}$ and $h:\mathbb{R}P^4\rightarrow S^1\times S^3$. Is there a lift ...
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206 views

Why a spiral is the deformation retract of a plane?

As the title says, why a spiral is the deformation retract of a plane?.
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324 views

Do we need net refinements not induced by preorder morphisms?

From Engelking's book on general topology (slightly rephrased): Definition: We say that the net $S': \Sigma' \to X$ is finer than the net $S: \Sigma \to X$ if 1. there exists a function $f: ...
2
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0answers
195 views

homework problem about the projective real space

Sorry for ask this problem, but I am very complicated with this problem :/ . My course it´s of topology, the teacher said that we only need the definition of the quotient topology and of $$ P_R^2 ...
2
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0answers
47 views

Set of 3D surfaces

How might one show that the set of connected 3D surfaces with infinite genus (up to homeomorphism) is countably infinite? I am guessing that we could either use proof by contradiction or come up ...
2
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0answers
273 views

Point-set topology and set theory

In the standard second-order, but single-sorted setting of point-set topology one has a base set $X$ and the property of being open on its powerset $P$ obeying the usual axioms. Proofs in point-set ...
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269 views

Generalized Jordan curve theorem (and a related MAIN QUESTION)

Preliminaries A Jordan map is a continuous map $f: [0,1] \rightarrow \mathbb{R}^2$ such that $f(0) = f(1)$ the restriction of $f\ $ to $[0,1)$ is injective A Jordan curve is a subset $\gamma$ of ...
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443 views

Properties of the universal cover of CW-complexes

Let $Y$ be a CW-complex and $X$ its universal cover. Could you give me a proof (or a referece) for the following fact: $X$ is contractible $\Leftrightarrow$ $H_i(X)=0$ $\forall i\geq2$ ...
2
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373 views

Learning analysis through topology

One of my supervisors once mentioned that when he was learning analysis he learnt it backwards. He learnt topology first and then saw analysis after, instead of the usual approach of doing everything ...
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288 views

Direct proof to show that a set is closed

I was trying to prove something, and I did it, but what I used is too exaggerated. The problem is: Let K be the cantor set, prove that the sets $$ \eqalign{ & \left\{ {\left| {x - y} ...
2
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265 views

uniform distribution on unit ball

If $S$ is a set of an countably infinite number of points uniformly distributed throughout the unit ball in $\mathbb R^n$, is there for every point $p$ in the ball and every real number $e>0$, a ...
2
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0answers
132 views

Property (T) for groups vs top

I have encountered two properties in different areas of math. One is the property (T) of groups and the other is the property (T) of topologies. What is the connection between these two ? Thank you. ...
2
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109 views

Contractions and Map Extensions

I'm going through Spanier and got stuck on the following problem: Show that a space $Y$ is contractible if and only if given a pair $(X,A)$ having the homotopy extension property with respect to $Y$, ...