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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Injectivity in the zero homology

I'm struggling with following step in an excercises about Mayer-Vietoris sequences: In one step the solution says this map is injective since $A \cap B$ is path-connected: $$ H_0(A \cap B) ...
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44 views

Homeomorphism between product of spheres and euclidean space

I need to prove that ${S^n} \times {S^k}$ is homeomorphic to a subspace of ${\mathbb{R}^{n + k + 1}}$ by constructing an explicit map between the two. I am unsure how to start this as I can't seem ...
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32 views

What simple topological properties of conic sections can be explored?

In the framework of my science fair project I am working on conic sections in different metric spaces. What simple topological properties/operations and so can I explore on them? Edit: To clarify, ...
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35 views

Proving open neighbourhood in topology

Let $X$ be the set $(\mathbb{R}\backslash \mathbb{N}) \cup \{1\}$. Define a function $f:\mathbb{R} \rightarrow X$ by $$ f(x) = \left\{ \begin{array}{ll} x & \mbox{if $x \in ...
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17 views

Weak uniform convergence

Let $(X,\|\cdot\|)$ a reflexive and separable Banach space, and note by $X^{*}$ its topological dual and $\omega$ its weak topology. Also, put $C_{\omega}(I,X)$ the space of the continuous mappings ...
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67 views

Can someone check my proof? (connectedness of real projective hyperquadrics)

Theorem: Let $Q_{\mathbb{R}} \subset \mathbb{P}^n_{\mathbb{R}}$ the set of real points of a projective hyperquadric. Prove that $Q_{\mathbb{R}}$ is connected with the topology induced by ...
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23 views

Does bounded and closed equal compact for sets of Borel probability measures?

Equip the space of Borel probability measures on a fixed closed subset X of the s-dimensional Euclidean space with the topology induced by weak convergence of probability measures. In this setting, ...
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45 views

Why is the punctured plane not homotopic to the circle?

I know that the fundamental group of $X = \mathbb R^2 \setminus \{(0,0)\}$ is the same as the fundamental group of the circle $Y = S^1$, namely $\mathbb Z$. However, $X$ and $Y$ are not homotopic, ...
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20 views

If $E ⊂ \mathbb{R}^n$ and $F ⊂ \mathbb{R}^m$ are both connected, why is $E × F$ connected in $\mathbb{R}^{n+m}$?

Question: Suppose that $E ⊂ \mathbb{R}^n$ and $F ⊂ \mathbb{R}^m$ are both connected. How can I show that $E × F$ is connected in $\mathbb{R}^{n+m}$. My Progress: Let $U ⊂ \mathbb{R}^n$ be open with ...
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14 views

Convolution of basis functions is a member of the same set of basis functions?

Suppose $\left\lbrace \Phi_i\right\rbrace_{i=0}^{\infty}$ is a complete basis of $\ell_1$. So if $M\in\ell_1$ we can write it as a linear combination of the basis functions $M=\sum_{i=0}^\infty ...
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20 views

Prob. 2, Sec. 18 in Munkres' TOPOLOGY, 2nd ed: A necessary and sufficient condition for the image of a limit point to be a limit point?

Here's Prob. 2, Sec. 18 in the book Topology by James R. Munkres, 2nd edition: Suppose that $f \colon X \to Y$ is continuous. If $x$ is a limit point of the subset $A$ of $X$, is it necessarily ...
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13 views

Munkres positive linear map definition of path product (page 328)

I'm confused by Munkres' definition of the path product using the positive linear map. He defines the positive linear map $p: [a,b] \rightarrow [c,d]$ to be the unique map of the form $x \mapsto mx + ...
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26 views

Uniform continuity with respect to parameter.

Let $\mathbb{X},\mathbb{Y}$ and $T$ metric spaces. A family $\{f_t\}_{t\in T}$ of maps $f_t:\mathbb{X}\to\mathbb{Y}$ is uniformly continuous with respect to parameter $t$ if, $$ (\forall ...
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33 views

Is uniform continuity a property of the category of completely regular spaces?

If $(X, U)$ and $(Y, V)$ are uniform spaces then one has the notion of a map $f : X \to Y$ to be uniformly continuous relative to $U$ and $V$. A uniform space $(X, U)$ induces a completely regular ...
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33 views

Subnet vs Subsequence

In topological space $X$, $X$ is compact iff every net has covergent subnet. It is true that a sequence is a net, so why we can't derive sequentially compact from compact?
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24 views

Interesting orderings open sets of a topology

Let X be a set of points an $\mathcal{Q}$ be a partition on X. The intuition I want to model is that X is a set of worlds considered possible by an agent and $\mathcal{Q}$ is a question whose ...
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28 views

Spectrum homeomorphic to $S^1$

Consider $$A=\mathbb{R}[X,Y]/(X^2+Y^2-1),$$ the quotient of the ring of polynomials in two variables, modulo the ideal generated by $f=X^2+Y^2-1$ of, equivalently, the ring of remainders modulo $f$. ...
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13 views

Geometric intuition for conjunctive spaces

A topological space $S$ will be called conjunctive if for each open set $A$ containing a point $p$, there's a point $q\in S$ satisfying $\overline{\left\{q \right\}}\subset A\cap \overline{\left\{p ...
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36 views

Which book to use in conjunction with Munkres' TOPOLOGY, 2nd edition?

Although Topology by James R. Munkres, 2nd edition, is a fairly easy read in itself, I would still like to know if there's any text (or set of notes available online) that is a particularly good ...
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25 views

One point compactification and field extension

We know that a topological space $X$ has a one point compactification if there exist a compact set $Y$ having $X$ as a subspace,where $Y\setminus X$ contains a single point Also for two such $Y$ and ...
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47 views

Möbius strip parameterization and charts

A parameterization of the möbius-strip is given by : $$\begin{align}M=\{ (x,y,z) \in \mathbb R^3: x &= \cos t(1+ s\cos(t/2)),\\ y &= \sin t(1+ s\cos(t/2)),\\ z &= s\sin(t/2), \\ t ...
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46 views

Map from $\mathbb{R}^3 \rightarrow \mathbb{R}^6$ is Immersion for…

$$\phi :\mathbb R^3 \rightarrow \mathbb R^6$$ $$(u,v,w)\rightarrow \phi(u,v,w)=(x_1,x_2,x_3,x_4,x_5,x_6)$$ where $ \quad x_1=u^2 \quad x_2=v^2 \quad x_3=w^2 \quad x_4=vw \quad x_5=uw \quad x_2=uv$ ...
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10 views

Unique function that gives an angle to a point of an interval.

(Sorry for my poor English in advance, as it is not my first language. Sorry for the vague title too, as I didn't know how to summarize the topic of the question). Here is an exercise that was ...
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20 views

Characterization of spaces for which some compactness properties are equivalent

Let $X$ be a topological space. I know of the following fact: If $X$ is pseudometrizable or (sequential and Lindelöf, e.g. second-countable) then $$ X \text{ is compact} \Leftrightarrow X \text{ is ...
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22 views

Induced representations of compact groups

I am taking a seminar that follows Serre's book "Linear Representations of Finite Groups", and I am preparing a talk on Chapter 7 on induced representations (Frobenius reciprocity, Mackey's formula ...
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35 views

Full Strength of Arzela-Ascoli

We define a family of functions $\mathcal{F}$ on a domain $\Omega$ to be equicontinuous if for each point $x \in \Omega$ and any $\epsilon > 0$, there is a $\delta_x > 0$ such that $|f(x) - ...
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37 views

Whitney and Tensor product structures on $BU$.

I have a question regarding the 2 H-space structures on $BU$. My current understanding (which may not be correct!) is detailed below. $BU$ admits two H-space structures described as follows: Let ...
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17 views

Equivalence relation on $S^2$ and unit square make same space

I think I understand a bit of this task, but I hope someone can look critical to my answers: Let $X$ be the space obtained from the sphere $S^2$ by gluing the north and the south pole (with the ...
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38 views

Let $H$ be a Hilbert space and $Φ≤H$ be equipped with a topology. Under which topology on $Φ^*$ is $H^*\ni f\mapsto\left.f\right|_Φ$ continuous?

Let $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space and $\left\|\;\cdot\;\right\|$ be the norm induced by $\langle\;\cdot\;,\;\cdot\;\rangle$ $\Phi$ be a vector subspace of $H$ equipped ...
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38 views

Intuition of open and closed sets?

A set $S$ is said to be open in a topological space $X$. If it doesn't contain its boundary and closed if it contains its boundary.is this intuition right,then how can open and closed set be depicted ...
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28 views

Reducible Heegaard splittings can be written as connected sums

Recall that a Heegaard splitting is a decomposition of a three manifold into a triple $(V,W,S)$ where V and W are solid handlebodies meeting along their common boundary S. A Heegaard splitting is ...
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39 views

Topology given by a relation

I have a problem with creating an equivalence relation ~ in a set : $ S^{1}\times S^{2}$ so that $ (S^{1}\times S^{2})/$~ (a quotient space of the given relation) is homeomorphic to 3-dimensional ...
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10 views

Criterion for relative compactness in uniform spaces

I am having problems in understanding a criterion for relative compactness given in a book (see below for details if you are interested) on SPDEs. However, I think it just invokes a pretty general ...
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33 views

Introduction to homology of simplitial complex.

I know the outline of homology theory of simplitial complex and be able to trianglulate some simple figures and compute the homology groups of them, but don't know theoretial details such as what kind ...
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18 views

Does weak Hausdorffication preserve equalizers and finite products?

There is a "weak Hausdorffication"-functor $wh$ from the category of compactly generated spaces (CG) to the category of compactly generated weak Hausdorff spaces (CGWH) given by quotienting out the ...
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48 views

how to evaluate homotopy group of this specific structure

I am a Ph.D. student of physics and now I have some problems regarding the evaluation of homotopy group of a specific structure. In a paper, a specific topological structure is defined. The structure ...
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44 views

Theta-space is a deformation retraction of the doubly-punctured plane, how to find equations.

That theta space is given by $S^1\cup(0\times[-1,1]) \subset\mathbb{R}^2$ it is said that this space is a deformation retract of the doubly punctured plane, here is the explanation I found: The ...
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35 views

Prove that if $f$ is continuous on a compact set then it is uniformly continuous

Prove that if $f$ is continuous on a compact set then it is uniformly continuous. Proof: Let $f:A\rightarrow \mathbb{R}$ be a continuous function and let $A$ be a compact subset in a metric ...
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44 views

Hausdorff dimension calculation of union of sets

$F$ is a Cantor set in $(0,1)$, $\dim_HF=1/5$. What's the $\dim_HE$ where $E=(F×R)\cup(R×F)$? By the product properties, I know that and $\dim_H(F×[0,1])=6/5=1+1/5$, which is the sum of hausdorff ...
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55 views

Proof check: Combination of convergent sequences in topological spaces

Let $\{x_n\}_{n=0}^{\infty}$ and $\{y_n\}_{n=0}^{\infty}$ be two convergent sequences in the topological space $\mathbb{R}$ , such that $$\lim\{x_n\}=x_0$$ and $$\lim\{y_n\}=y_0$$ Show that: a) ...
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If $G$ is a compact Lie group acting effectively on $X$ then it is a subspace of Homeo$(X)$?

Let $G$ be a compact Lie group acting effectively on a simply connected space $X$. Let Homeo$(X)$ be the group of all homeomorphisms of $X$ with itself given the compact open topology. Is the ...
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27 views

Few complete subspaces of the space $\mathbb R^n$

$$X\subset \mathbb R^n$$ a subspace with Euclidean metric on $\mathbb R^n.$ Then the $X$ that is complete is : $A.X=\mathbb Z \times \mathbb Z\subset \mathbb R\times \mathbb R.$ $B.X=\mathbb ...
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Need a decent book covering Grassmannians and surrounding theory

As the title states, I'm looking for a decent book (advanced undergraduate, or graduate level) covering Grassmannians and surrounding theory. It seems to be that Grassmannians are hiding in a great ...
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24 views

$S^3$ viewed as a subspace of $\mathbb{C}^2$ and the solid torus, embedding

Consider the 3-sphere $S^3$ viewed as a subspace of $\mathbb{C}^2$: $$S^3=\{(u,v):u,v\in\mathbb{C}, |u|^2+|v|^2=1\}.$$ Inside the sphere we consider $$A:=\{(u,v)\in S^3: |v|=\frac{\sqrt{2}}{2}\}.$$ I ...
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25 views

Poincare Triangles given Vertices

If I'm given 3 vertices, say (5,3), (9,3), and (9,7), how can I draw this triangle in a 2 dimensional way along 2 axes and still represent it as a Poincare triangle?
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42 views

T/F: The set: $\{(x,y): \sin(x^{2012} +y^3) + x^2 + y^4 \le 1\}$ is a compact set in $\mathbb{R}^2$.

T/F: The set: $\{(x,y): \sin(x^{2012} +y^3) + x^2 + y^4 \le 1\}$ is a compact set in $\mathbb{R}^2$. I think it is false since I don't believe the set is closed nor bounded? Is that correct?
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33 views

Alternate proof that continuous function on a compact set is bounded and attains a maximum

I have already proved that a continuous real-valued function maps compact sets into compact sets. So a continuous function on a compact set is bounded and attains maximum. I want to prove these facts ...
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35 views

Limit point of a sequence

Given a sequence $(a_n)_{n \in \mathbb{N}} \subset \mathbb{R}$ and a set $M = \{a_n : n \in \mathbb{N}\}$, is $|M| < \infty$, then $(a_n)_{n \in \mathbb{N}} $ has a limit point. I believe that ...
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43 views

Proof verification: if $E$ is a finite dimensional subspace of a normed space $X$, then $E$ is a closed subspace.

Could someone please verify the following proof for this statement: If $E$ is a finite dimensional subspace of a normed space $X$, then $E$ is a closed subspace. Proof: If $E$ is a finite ...
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22 views

Continuous real valued function on a hausdorff space whose support is contained inside an open set

Yet another question as part of this whole mess. Context: I'm trying to show that for a compact Hausdorff space X, the collection of sets $U_f=\{x\in X:f(x)\ne 0\}$ form a basis, where $f$ ranges ...