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 vector space with topology generated by a family of typologies each makes it a topological vector space is a topological vector space

Let $V$ be a vector space, and let $(\mathcal F_ \alpha ) _{ \alpha \in A}$ be a family of topologies on V, each of which turning $V$ into a topological vector space. Let $\mathcal F$ be the vector ...
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36 views

Contractibility of $ S^2 $

In my lecture notes, it said $ S^2 $ was not contractible. If I think about $S^2$ as balls that are shrinking I can see that it is not contractible because the balls of smaller radii area not "in" ...
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28 views

Sequence Criterion of Continuity and Divergent sequences

A function $f : (X, d_X) \to (Y, d_Y)$ between metric spaces is continuous iff $$ \lim_{n \to \infty} f(x_n) = f(\lim_{n\to \infty} x_n) $$ for each convergent sequence $(x_n)$ in $(X, d_X)$. So if I ...
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20 views

On the Baire spaces

In our paper on arXiv:1403.1529 [math.GN] , we provide some conditions under which we ensure that a subspace is not Baire. We proposed an example that is not known for us that is there another ...
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31 views

Learning about Calabi–Yau manifold

I want to Create a 3d animation of Calabi–Yau manifold. I tried to learn it but it deals with some very advanced math. What math knowledge have to know before trying to understand the calabi-yau ...
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25 views

Is the set, $\{f\in Y: \left\|f\right\|_{Y} \leq C \ \text{and} \ \left\|(|f|)\right\|_{X}\leq C \}$, closed in $(Y, ||\cdot||_{Y})$?

Put, $X= \text{The space of "nice" complex valued functions on } \mathbb R; \text{that is}, f:\mathbb R \to \mathbb C;$ so that $X$ is Banach space with respect to the norm ...
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13 views

What is the topology of pointwise convergence of increasing functions?

Given a set $X$ and a topological space $Y$, the topology of pointwise convergence on the set of all functions from $X$ to $Y$ is the product topology on $\Pi Y_x$, since by definition, this is the ...
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47 views

Continuity of a function that maps non convergent sequences onto non convergent sequences

Let be f a surjective real function defined on R mapping every non convergent sequence onto non convergent sequence. Prove that f is continuous. I have proved that f is injective. Can you help me ...
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16 views

Whats the difference between a barrelled space and a locally convex one?

Wikipedia says that a locally convex space is a topological vector space whose topology is generated by translations of balanced, absorbent, convex sets. Whereas a barrelled space is one where: ...
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14 views

Projective topology (reference)

Please suggest me a good reference to study Projective topologies. I just want an introductory exposition.
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27 views

A dense subset of a finite group

Let $G$ be a finite group with Zariski topology. Suppose $G=A_1\cup A_2\cup\cdots\cup A_n$, where $A_i$, $1\leq i\leq n$, are pairwise disjoint subsets of $G$ and only $A_1$ is dense in $G$, that is, ...
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29 views

terminology: accumulation points, limit point, cluster point

In a topological space $X$, what would be the most common terms to describe the following two properties about a point $x\in X$ and a subset $S\subseteq X$. I) For every open set $U$ with $x\in U$, ...
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36 views

X is a compact topological space?

Let $X$ be a topological space. Then, when it is said that $K \subset X$ is compact, it is clear to me that every open cover of $K$ contains finite subcover of $K$. But what do we mean when it is ...
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17 views

Can the Hjalmar Ekdal topology be defined on uncountable sets?

Can the Hjalmar Ekdal Topology be defined on uncountable sets and how would the various topological properties change from those associated with the set of positive integers? (Example 55 in ...
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28 views

What kind of norm is it in the definition of $S^{n-1}$?

Definition (wikipedia) $S^n\triangleq\{x\in\mathbb{R}^{n+1}: ||x||=1\}$ is said to be a 'n-sphere' What norm is it referring to ? I have proven that ...
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71 views

Totally bounded uniform spaces vs proximity spaces (need proof)

nLab says "The category of totally bounded uniform spaces and uniformly continuous functions is equivalent to the category of proximity spaces and proximally continuous functions". How to prove this? ...
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21 views

Cauchy filters defined for proximity spaces?

I define in my draft article Cauchy filters $\mathcal{X}$ on a uniform space $\nu$ by the formula: $$\mathcal{X}\ne\bot \wedge \mathcal{X}\times^{\mathsf{RLD}}\mathcal{X}\sqsubseteq\nu.$$ ...
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51 views

Show that $E= \{1, 1/2, 1/3, \dots\} \cup \{0\}$ is compact

Here is my answer hope it makes sense Let the open cover $\{\theta_i\}, i \in I$ be given. Every element of E belongs to $\cup_{i \in I} \theta_i$. Choose $i_0 \in I$ such that $0 \in \theta_{i_0}$ ...
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28 views

Show that the cantor set is self similar

Alright so here is what I have: $$\sum_{n=0}^\infty \frac{1}{3}\frac{2}{3}^{n-1} = \frac{1}{3}\sum_{k=0}^\infty\frac{2}{3}^{k}$$ $$\sum_{n=0}^\infty r^{k} = \frac{1}{1-r}, r\lt 1$$ ...
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20 views

Compactly Supported Partitions of Unity

What conditions on a locally-compact Hausdorff topological space $X$ does one need to be assured the existence of a partition of unity on $X$ so that each function in the partition of unity has ...
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35 views

Does every bounded Lebesgue measurable set of non-zero measure contain a boxed compact set?

Does every bounded Lebesgue measurable set $A$ (of nonzero measure) in $\mathbb{R}^N$ contains a compact set of the form $$I_1 \times I_2 \times \dots \times I_N$$ where $I_i$s are finite closed ...
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19 views

Minimal Cauchy filters in Cauchy spaces

It is well known that "every Cauchy filter contains a unique minimal Cauchy filter" (Wikipedia) for both metric spaces and uniform spaces (see also this question and answer). Does this theorem ...
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47 views

Homeomorphism between the pullback space of a map and the disjoint union of two unit intervals.

Given a path $\beta: I\to \mathbb{RP}^1; t\mapsto p(e^{2\pi i t})$, where $p: S^n\to \mathbb{RP}^1$ is the canonical projection I want to show that the pullback space $$ \beta^* S^1=\{(t,y)\in I\times ...
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38 views

How to prove (S1, *) is a topological group?

I'm trying to prove that (S1, *) is a topological group. Where S1 = {z complex : |z| = 1}. So I want to show that S1 x S1 -> S1: (z,w) -> z*w and S1 -> S1 (z) = z^(-1). I'm not really sure where to ...
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31 views

A homeomorphism $\overline{k} \otimes_k K \rightarrow \bigoplus K_j$

Let $k$ be a field with a non-archimedean absolute value $||$, $K/k$ a separable extension of degree $N$. Also let $K = k(\beta)$, and $\mu$ the minimal polynomial of $\beta$ over $k$. If ...
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22 views

Clarifying the notion fo fiberwise cone

I want to understand the notion of "fiberwise cone". The only "definition" I found is the following (Ichirō Satake, University of Toronto Press, 1991 ): There is a more general construction. ...
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27 views

Product of manifolds

Suppose that $M_i$ is a manifold with atlas $\{(U_{ij}, f_{ij})\}$ for $i=1,2,\ldots,k$. Then we note that $M_1\times M_2\times\cdots\times M_k$ is a manifold with atlas $\{(U_{ij}\times U_{sj}, ...
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13 views

Extendability of Contact Structures, Foliations of S^2

I am currently reading Eliashberg's paper on the classification of overtwisted contact structures (http://bogomolov-lab.ru/G-sem/eliashberg-tight-overtwisted.pdf). In it, there is the following ...
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22 views

Bin Packing Algorithm and Minimal Area Axis-Aligned Bounding Boxes

I am a computer hobbyist and, just for the heck of it, I have decide to work on a bin packing algorithm. I would like for the program to eventually handle complex 2-D objects with bezier curves and ...
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42 views

Proving boundary set inclusion

Let there be $D_1$, $D_2 \subset \mathbb{R}^{m}$ I want to show that if $D_1 \nsubseteq \nsupseteq D_2 \wedge D_1 \cap D_2 \neq \emptyset$ then $\partial (D_1 \cap D_2)\subset (\partial D_1) \cup ...
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12 views

What are topological relationships? Why are they invariant to transformations such as scaling, translating and rotation?

The following paragraphs are from this paper: When computing the spatial similarity between two pairs of regions, we consider that all the basic relations should not contribute in the same ...
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47 views

Basic Topology Proof

Let $\mathscr{E} \subset \mathscr{P}(X)$ for some set $X$, let $\mathscr{T}_{\mathscr{E}}$ be the smallest topology containing $\mathscr{E}$. Constructed in the following manner: Let $A \in ...
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17 views

Proving that products of paths respects homotopy rel endpoints

I'd like to show that given paths $f \cong g$ and $f' \cong g'$, homotopic relative their endpoints, then $f \cdot f' \cong g \cdot g'$ assuming of course that the $f$ ends where $f'$ begins, and ...
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29 views

The space Thomas's Planck from “counterexamples in topology”

Anyone knows where can I get a description of the space " Thomas's Planck "? It is mentioned as example 93 in the book "counterexamples in topology" but there is no description of that space there. ...
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25 views

Hierarchy of Topologies

I'm looking over some questions, studying for my topology midterm tomorrow and I've come across a question I'm unsure about: Show that the $T_1$-topology on the real line is coarser than the ...
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44 views

Generating an open set using basis of standard topology.

Let $(\mathbb R, \mathscr T$) be a topological space where $\mathscr T$ be a standard topology. Let $K = \{ \frac1n | \; n \in \mathbb N \}$. How can I generate $K$ from the basis elements of this ...
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10 views

How can one charactise the topology for sequentially closed sets for increasing sequences?

When looking at lesbegue integration one often comes up against increasing sequences of functions that are convergent in a space. So, abstracting: given a poset $(X,\leq)$ is there a nice ...
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22 views

Topology generated by a uniformity

Let $\mathfrak{X}$ be the complete lattice of all filters (including the improper filter) on $U\times U$ (for some set $U$), with the order being the set inclusion of the filters. Consider the ...
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28 views

Can this collection of sets be the basis for a topology?

A topology $\mathfrak{T_5}$ is defined in Munkres to have the following basis: $(-\infty,a)$, where $a\in \Bbb{R}$. How can this be a basis? Say $B_1$ and $B_2$ are two basis sets. Then there should ...
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35 views

Proofs involving closedness of compositions of mappings.

I've feel like I've gotten myself in over my head in a non-linear optimization course I seem to lack the mathematical maturity for(I'm an undergrad, I've taken the calc series, Intro to differential ...
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26 views

Prove: If a finite dimensional vector space $V$ is bounded by a norm $p$ then $V$ is bounded by any norm.

Prove: If a finite dimensional vector space $V$ is bounded by a norm $p$ then $V$ is bounded by any norm. Norms on a finite-dimensional vector space are all equivalent meaning they can be bounded ...
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36 views

Topological Space Intuition

I gained some insight recently when I read an intuitive explanation for what "rich" topologies actually model. It said, basically, that a rich enough topology (say, a Hausdorff one) is a model for ...
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12 views

Countable product of metric spaces

Let $X=\prod X_i$ of countably many metric spaces $(X_i,d_i)$. Prove that the function which associates to $x=(x_i)$,$y=(y_i) \in \prod X_i$ the number $d(x,y)\in [0,\infty]$ defined by ...
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33 views

Confusion in an order topology being Hausdorff and T1.

There is a theorem which says that every order topology is Hausdorff. Also every Hausdorff follows $T_1$ Axiom. So suppose $X = \{1,2\}$. Now $ 1<2$. The bases for the topological space $X$ can be ...
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33 views

seperated sets and Convex subsets

Let $A, B ⊂ \mathbb{R}^n$ be separated subsets. Pick $p, q $ in $A, B$ respectively and define $l(t) = −1 −1(1 − t)p + tq $ for $t ∈ \mathbb{R}$ . Show that the sets $l^{-1}(A), l^{-1}(B)$ are ...
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31 views

A Nonseparable Topology of a Set

Let a set $X$ have topology $T$. Let the topology be the trivial topology. Let $T$ also be nonseparable. What does this imply for $X$?
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29 views

$x$ is an accumulation point iff $x$ is in the closure of $A\backslash\{x\}$

I'm given $X$ is a Hausdorff topological space, and $A$ is subspace. I'm trying to show that $x$ is an accumulation point of $A$ iff $x$ is an element of the closure of $A\backslash\{x\}$. So far I ...
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18 views

Why antipodal S2 mapping has degree -1 and why this map doesn't?

Hi I'm reading the post here (Shortest proof for 'hairy ball' theorem) by Arthur to explain that there is no smooth vector field on S2. I don't understand it very well: The simplest I can ...
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49 views

Chart on a manifold

I have the following question. If I consider a manifold, for example a torus T see as space of identification $[0,1]\times [0,1]$ why I can't cover it with only one chart? what fails if a chart ...
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32 views

Proof verification and suggestion to elude the AC (equivalent definition of adherent points).

Hi everyone I'd like to know if the following is correct and, more importantly, if there is some way to escape of the axiom of choice (as the hint the book says "use AC"). Definition: Let $X\subset ...