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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28 views

a question about Fenchel's theorem(differential geometry)

I am an undergraduate student studying differential geometry right now. I am just finishing reading how to prove Fenchel's theorem:The total curvature of a smooth closed curve in 3-dimensional space ...
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40 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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15 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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30 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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37 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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46 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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41 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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18 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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21 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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33 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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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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35 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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35 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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43 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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13 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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21 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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26 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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37 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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90 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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40 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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13 views

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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19 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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20 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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25 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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51 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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39 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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23 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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28 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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27 views

Prob. 7, Sec. 26 in Munkres' TOPOLOGY, 2nd ed: How is the projection onto the first factor closed in 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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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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51 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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34 views

Theorem to show trajectories of differential equations are close after small change to initial condition

Consider two solutions(or trajectories), say $x_1(t)$ and $x_2(t)$, of a system of differential equaions. That is, $$ x_1'(t)=x_2'(t)=f(x,t), t\ge0. $$ Also, $\|x_2(0)-x_1(0)\|<\epsilon$ for some ...
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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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37 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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29 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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23 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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80 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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30 views

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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22 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 ...