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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2
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64 views

Inverse limits and sums

Let $(X_\alpha)_{\alpha<\omega_1}$ be a family of compact metric spaces such that $X_\alpha$ is homeomorphic to a subspace of $X_\beta$ for $\alpha<\beta$. Can we regard the disjoint sum $\...
2
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133 views

Quotient map from the $(2n+1)$-dimensional sphere into complex projective space is open.

We have a natural quotient map $$\phi\colon S^{2n+1}\rightarrow (\mathbb{C}^{n+1}\setminus{\{0\}})/\mathbb{C}^{*}=\mathbb{P}^n\mathbb{C}$$ and I want to see, that it's open. Denote by $$\iota\colon S^{...
2
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139 views

Homeomorphism between simply connected, closed 3 - manifold and 3-sphere.

The Poincare conjecture states that a simply-connected, closed 3-manifold is homeomorphic to the 3-sphere. Now that the conjecture has been settled, could someone tell me what this homeomorphism is (...
2
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289 views

topology-cofinite topolgy and co-countable topolgy

Let $T_1=\{U:X-U \text{ is finite for all of } X\}$. Then $T_1$ is the cofinite topolgy on $X$,where $X$ is an arbitrary infinite set. Then $T_1$ is not a Hausdorff space.Is it a regular space or a ...
2
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42 views

Understanding intuitively that any loop $p:[0,1] \to S^1$ is end point preserving homotopic to a loop which doesn't change direction.

I'm trying to understand intuitively the proposition that: Any loop $p:[0,1] \to S^1$ is end point preserving homotopic to a loop which doesn't change direction. Surely a loop round a circle ...
2
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0answers
286 views

Properly discontinuous action on a non-locally compact space

Let me begin with some definitions in order to avoid confusion. An action of a group $G$ on a space $X$ is proper if the map $G \times X \to X \times X$ given by $(g, x) \mapsto (x, gx)$ is proper, i....
2
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67 views

How to prove an isotopy relative to a point exist?

Let $M$ $ $ be a differential manifold, and $f$ a diffeomorphism on $M$ which is isotopic to $id$. Assuming that $x\in M$ is a fixed point of $f$ and the orbit of $x$ under the isotopy is a trivial ...
2
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155 views

Affine transformation

Let $S_1$ and $S_2$ be sets. Let $n_1$ be the cardinality of $S_1$ and $n_2$ be the cardinality of $S_2$. I assume that $n_1$ and $n_2$ are finite. Let $e$ be a function that maps members of $S_1$ and ...
2
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150 views

A Cartesian product of metric spaces is perfectly $\kappa$-normal

A space $X$ is called perfectly $\kappa$-normal if the closure of any open set (that is, every canonical closed set) is a zero-set. How can i prove this proposition directly? $Proposition$: A ...
2
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185 views

F-space and completely regular space

A completely regular space X is an F-space if for each functionally open set $M\subset X$ every continous function $f: M \rightarrow I $ continuously extendable over X. I want to prove that a ...
2
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20 views

Edwards-Anderson Hamiltonian of a Hopf link

I was calculating the Edwards-Anderson Hamiltonian of a Hopf link. A hopf link is like attachment 1. I have drawn the Seifert surface of that link. The surface is shown in attachment 2. It also ...
2
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119 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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68 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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202 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 ...
2
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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 $U$ ...
2
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101 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 ...
2
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113 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. ...
2
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42 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 ...
2
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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 ...
2
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122 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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193 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 ...
2
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0answers
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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67 views

Pointfree generalization of uniform spaces?

Topological spaces generalize as frames and locales. But are there a pointfree generalization of uniform spaces?
2
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144 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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0answers
177 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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0answers
491 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 $C^0(\...
2
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0answers
52 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 ...
2
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0answers
394 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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69 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 continuous?...
2
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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 ...
2
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148 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 ...
2
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0answers
198 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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0answers
246 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 ...
2
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111 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 $\mathbb{A}^{1,an}_{^\rho \mathbb{R}}$...
2
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0answers
941 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 ...
2
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0answers
89 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
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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0answers
612 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 $S^...
2
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0answers
114 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 $(S^2\...
2
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0answers
288 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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0answers
354 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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0answers
97 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, ...
2
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0answers
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 $\mathbb{P}_{2}(\...
2
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0answers
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 situation:...
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 $g:\mathbb{R}P^4\...
2
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0answers
210 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?.
2
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0answers
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
197 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
280 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 ...