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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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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40 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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38 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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37 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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15 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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24 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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29 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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46 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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98 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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45 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 ...
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41 views

technique of showing a set is a topology

Let $X $ be a set and let $Y\subset X$. Define $\tau_Y $ to be the collection of all subsets U of X such that $Y\subset U $ or $U= \emptyset $ . Prove that $\tau_Y $ is a topology on $X $. I have ...
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Are there any other measures of complexity for a continuous map than topological entropy?

Let $X$ be a compact topological space and $T\colon X\to X$ be continuous. In order to say something about the complexity of $(X,T)$ there is of course the notion of topological entropy of $T$, ...
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21 views

Upper-hemicontinuity of product maps on compact metric spaces.

Let $X$ and $\{Y_i\}_{i\in I}$ be compact metric spaces (where $I$ an index set of possibly uncountable cardinality). Let $\Gamma_i$ be a compact valued, upper hemicontinuous (UHC) correspondence from ...
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29 views

Zips and Zippers

I'm currently reading Differential Manifolds by Antoni Kosinski, and the concept of a zip--defined as half of a zipper--is mentioned early on, of course with the intent of connecting manifolds. This ...
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21 views

Disjoint Union of Completely Regular Spaces

I am trying a new approach to an already-solved problem, but I need help to see if I'm on point. Munkres Chapter 53, question 6 [abridged] asks, given a covering map $p: E \to B$: Show that "if $B$ ...
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27 views

On the covering dimension of an image under a continuous function

I'm trying to solve the following exercise: Let $X$ be a compact Hausdorff space and let $U_1,...,U_n$ be a cover of $X$ of order $m$. Let $z_1,...,z_n\in\mathbb{R}^N$ for some $N$ be in general ...
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30 views

Showing any metric space is a Hausdorff space

This is the question i've been given along with the solution i have written, however, could someone explain why showing $z \notin V$ shows any metric space is a Hausdorff space
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17 views

Continuity of Component Function

Let $f:Z\times X \to Y$ be given such that $f$ is continuous. I'm trying to prove that $f(z, -)$ is continuous for a fixed $z\in Z$. I would appreciate if someone could tell me if the proof that ...
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52 views

Projection of a covering map over product set.

Let $p,q$ be a covering maps, $p:\tilde X \rightarrow X$ and $q:\tilde Y \rightarrow X$ and let $Z=\lbrace(\tilde x, \tilde y)| p(\tilde x)=q(\tilde y) \rbrace$, I want to proff that $f:\tilde ...
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40 views

Topologically equivalent metrics, using different definitions.

I´ve been dealing with topologically equivalent metrics for a while, using the usual definition, that $d$ and $d'$ are topologically equivalent iff they have the same open sets. However, there is ...
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24 views

Topological Embedding Which is Neither Open nor Closed

I'm having trouble coming up with an example of an embedding which is neither open nor closed. My attempts have included trying to find such a map from $\mathbb{R}$ (given the usual Euclidean ...
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29 views

core-compact but not locally compact

A space $X$ is called core-compact if the set of all open set in $X, \mathcal{O}(X)$, is a continuous poset. It is known that every locally compact is core-compact. Here, a space $X$ is locally ...
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37 views

Prob. 7, Sec. 26 in Munkres' TOPOLOGY, 2nd ed: How is the projection onto the first factor closed if the second factor is compact?

Let $X$ and $Y$ be topological spaces such that $Y$ is compact. Then how to show that the projection map $\pi_1 \colon X \times Y \to X$ is a closed map? My effort: Let $C$ be a non-empty closed ...
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145 views

Fubini's theorem on a product of locally compact spaces which do not have countable bases

Let $X$ be a locally compact Hausdorff space. Let $\mathcal B$ be the $\sigma$-algebra generated by the family of open subsets of $X$. A measure $\mu$ on $\mathcal B$ is called a (positive) Radon ...
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21 views

A sequence with two distinct limits

I just wanted to check I was right about this: Consider $X=\{1,2,3\}$ equipped with the topology $T=\{\emptyset,\{1,2\},X\}.$ Then the sequence $(1,2,1,2,1,2,\ldots)$ converges to both $1$ and $2$ ...
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24 views

How can I show uniqueness of a (constructed) topology.

Let $X$ be a set, and $\Phi$ a set of subsets of $X$ such that: $\varnothing, X \in \Phi$ If $\{ F_i: i\in I\}\subseteq \Phi$, then $\bigcap_{i} F_i \in \Phi$ If $F,G \in \Phi$ then ...
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53 views

Almost complex structure gluing

Consider smooth complex manifold $X$ of complex dimension $n$ and its smooth submanifold $Z$ of codimension $k$. Denote the normal bundle of the inclusion $Z\subset X$ by $\nu.$ One can blow up $X$ ...
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32 views

Is imaged of a Polish Hausdorff space under an injecitve map always Hausdorff?

I have a question about Hausdorff topological space. Question: Let $X,Y$ be topological spaces. If $X$ is a Polish space (i.e. $X$ is a separable and completely metrizable space.) and $Y=f(X)$ ...
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25 views

Prove that a point $(a,b)$ in $\mathbb{R^2}$ has the same homotopy type as $\mathbb{R^2}$.

Prove that a point $(a,b)$ in $\mathbb{R^2}$ has the same homotopy type as $\mathbb{R^2}$. If someone could verify my proof that would be great. I just started this learning this material and I ...
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38 views

Is Hausdorffness characterisable by the uniqueness of the limits?

It is clear that if the points $x$ and $y$ are separated by neighbourhoods, then there is no sequence which converges to $x$ and to $y$ as well. But when I try to prove that if $x$ and $y$ aren't ...
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31 views

Topological group, which is second category in itself, is a Baire space.

A Baire space is a topological space in which the union of every countable collection of closed sets with empty interior has empty interior. $G$ is a topological group, if $G$ is of the second ...
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24 views

Interior/boundary of unit diagonal

Trying to wrap my head around interiors and boundaries of subsets. I'm attempting to find the interiors and boundaries of the open and closed unit diagonals w.r.t the product topology on ...
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89 views

The skeleton of Eulerian polyhedra

There is (at least) two kind of validity domain of Euler's $v−e+f=2$ polyhedron formula. One is the "Eulerian" polyhedra, i.e. simply connected polyhedra with simply connected faces (see here). The ...
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Prob. 5, Sec. 25 in Munkres' TOPOLOGY, 2nd ed: Is there a connected set that is locally connected at none of its points?

Let $A$ denote the rational points of the interval $[0,1] \times 0$ of $\mathbb{R}^2$. Let $T$ denote the union of all line segments joining the point $p = 0 \times 1$ to points of $A$. Then I can ...
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38 views

A problem about the intersection of convex open sets in $\Bbb R^m$

Let $X\subset \Bbb R^m$ be the union of convex open sets $X_1,\cdots,X_n$ such that $X_i\cap X_j\cap X_k\neq\varnothing$ for all $i, j, k$. Is $\bigcap\limits_{r=1}^nX_r\neq\varnothing$ true?
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23 views

Basis for a topology of a scheme

Suppose that $X$ is a proper and connected scheme over an algebraically closed field. Moreover let $\mathcal A$ be a collection of open subsets of $X$ with the following property: For every open ...
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29 views

Let $X=\mathbb Z^+$. Topologize $X$ with the finite complement topology. Let $A=\{3,5,6,7\}$. Find $\overline A$, int$(A)$, and bdry$(A)$.

Let $X=\mathbb Z^+$. Topologize $X$ with the finite complement topology. Let $A=\{3,5,6,7\}$. Find closure of $A$ $(\overline A)$, interior of $A$ (int$(A)$), and boundary of $A$ (bdry$(A)$). $A$ ...
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39 views

Shortest smooth paper Möbius Strip

I want to make a familiar Möbius strip of width 1 unit satisfying the physical properties of paper. Assume paper is a ruled surface, and the strip has to be smooth and non-self-intersecting. What ...
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32 views

A name for a particular covering map?

The quotient space of $\mathbb C$ obtained by identifying points differing by a Gaussian integer is topologically a torus. The map that takes each point in $\mathbb C$ to its corresponding point in ...
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73 views

Does the proof of productivity of connectedness require Axiom of Choice?

For arbitrary index set $\Lambda$, the product space $$ X = \prod_{\alpha \in \Lambda} X_{\alpha} $$ with product topology is connected if all of each $X_{\alpha}$ is connected. In the standard proof ...
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31 views

Is the boundary of a set a subset of the limit points?

Let $(X, \mathfrak T)$ be topological space and suppose that $A$ is a subset of $X$. Then $Bd(A) \subseteq A'$. My definition of boundary: Let $(X,\mathfrak T)$ be a topological space and let $A ...
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44 views

Path-components of the general linear group using only elementary algebra

Let $E(c)$ be an elementary matrix of the type to add $c$ times a row to another row when applied to another matrix on the left (with $c$ in some off-diagonal position $(i, j)$), and, with the usual ...
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Prove that the “additive” operation of the module($\mathbb{Z}_{p}^{*},\mathbb{Z}_{p-1},\cdot$) is continuous.

Consider the following module $\mathcal{M}=$($\mathbb{Z}_{p}^{*},\mathbb{Z}_{p-1},\cdot$) in which the "additive" operation is defined by normal multiplication in $\mathbb{Z}_{p}^{*}$ and scalar ...
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18 views

Finite intersection of arbitrary union not stable for arbitrary unions

It is a set-theoretic exercise to prove that the set of arbitrary unions of finite intersections of sets is still stable under finite intersections. However it is not true that finite intersection of ...
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33 views

weak closure of unitary group in $B(H)$

Let $H$ be a Hilbert space with dim $H=\infty$ , and $\cal{U}$ be the group of all unitaries on $H$. Show that the weak closure of $\cal{U}$ is a semigroup with respect to the multiplication. I know ...