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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16 views

Topology induced by quasi-pseudo-metrics

So I know that uniform spaces can be described using a collection of metrics which satisfy 2 properties. For a topological space $(X,\mathcal{T})$ you can define a collection of pseudo-quasi metrics ...
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13 views

Are trace of z-filter in dense z-embedded subset z-filter?

I found this article about z-filter, referring to Lemma 3 my question is: without the "every member of which meet Y" hypothesis and adding that Y has to be dense in X is it still true? EDIT: forgot ...
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38 views

Plotting Distance Constrained Points on a Plane

Does anybody know of some algorithmic way to tell if it is possible to plot a set of distance constrained points on a cartesian plane. Or, better still, a method to determine the minimum number of ...
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25 views

Prob. 1, Sec. 28 in Munkres' TOPOLOGY, 2nd ed: An infinite subset of $[0,1]^\omega$ without limit points in the uniform topology?

Let $[0,1]^\omega$ denote the set of all sequences of real numbers in the closed unit interval $[0,1]$, and let the uniform metric $d$ on $[0,1]^\omega$ be given by $$d\left( (x_n)_{n\in\mathbb{N}} , ...
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49 views

Confusion regarding continuous functions between topological spaces – a subtle but possibly important point

Let $T: V_1 \to V_2$ be a linear mapping. Show that $T$ is a continuous function between $(V_1, \tau_{V_1}) $ and $(V_2, \tau_{V_2}) $ A direct solution to the problem is not what I am looking ...
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40 views

The space of continuous functions as a dual space

Let $X$ be some topological Hausdorff space and $C_b(X)$ the space of bounded complex continuous functions on $X$. Is there a Banach space $B$ such that $B^* \simeq C_b (X)$? I know of a very similar ...
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35 views

Separated subsets of $\mathbb{R}^k$

Let $A$ and $B$ be separated subsets of some $\mathbb{R}^n$, suppose $a\in A, b\in B,$ and define $p(t)=(1-t)a+tb$ for $t\in \mathbb{R}^1$. Put $A_0=p^{-1}(A), B_0=p^{-1}(B)$. (a) Prove that $A_0$ ...
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37 views

n-dimensional lattice as a collection of lower dimensional spaces.

Suppose I have n orthogonal unit vectors (in Euclidean space). These unit vectors may be used to describe an n-dimensional unit hypercube. A subset of these unit vectors may be combined to make lower ...
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27 views

Peano space - A compact connected locally connected metric space

Let $X$ be a peano space. Is it possible to find a countable collection $\mathcal{\mathbb{B}}$ which forms a base for $X$ and every member of $\mathcal{\mathbb{B}}$ is a peano subspace of $X$.
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13 views

Question regarding characters and point open topology2

this is a follow-up question for the following one: Dual group of G with point open topology is an intersection of C(G,T) and a closed set In the book of Banaszczyk - "Additive subgroups of ...
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20 views

Lemma in Farkas/Kra: Riemann surfaces on construction of domain satisfying certain properties

I'm having some trouble understanding the proof of this lemma. I can follow the construction of $u$ and $D$, however the final step in the proof seems to be without justification. Specifically why ...
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28 views

Order topology on a poset

How does the open rays on a partially ordered set X forms a subbasis for a topology on X (which is called order topology) ? I was considering the case X has only one element. Moreover, I know if the ...
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20 views

What quantities does a local topological region have in 3D?

If we take an infinite solid R3 and cut out a torus and sew it back in with Dehn surgery. This will create a local topological region in R3. I was thinking.. are there any characteristic values ...
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49 views

Are Prevarieties irreducible?

In Goertz-Wedhorn, a prevariety is defined to be a connected space with functions that locally is an affine variety (were an affine variety is a space with functions that is isomorphic to the space ...
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27 views

Is the coset space $\frac{SO(3)\times Z_2}{H \times Z_2}$ isomorphic to $\frac{SO(3)}{H}$?

I have heard many times that the homotopy group of the coset space $\frac{SO(3)\times Z_2}{H \times Z_2}$ and of the space $\frac{SO(3)}{H}$ are identical. I.e., $\frac{SO(3) \times Z_2}{H \times Z_2} ...
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27 views

Is the subspace generated by complete othogonal subspaces closed?

$E$ is a vectorial space equipped with an inner product $\langle \cdot, \cdot \rangle$. $(E_i)_{i \in I}$ is a family of complete pairwise orthogonal subspaces. Is the subspace $V$ generated by the ...
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56 views

Homeomorphic to $\mathbb{R}^n$

We know that $D^n = \{x = (x_1,x_2,\ldots,x_n)\in\mathbb{R}^n:\|x\|<1\} $ homeomorphic to $S^n_+ = \{x = (x_1,x_2,\ldots,x_n)\in\mathbb{R}^{n+1}:\|x\|=1, x_{n+1}\geqslant 0\}$ and homeomorphic to ...
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76 views

Homeomorphisms in $\mathbb{R}^2$

Consider $(0,0)$ in $\mathbb{R}^2$. Is there a neighborhood $U$ of $(0,0)$ open in the upper-right quarter plane but not in $\mathbb{R}^2$ (I mean such that if $(x,y)\in U$ then $x,y\ge0$) which is ...
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43 views

How to understand this sentence within Atiyah-Macdonald's textbook about commutative algebra

In page 102 of this textbook, authors mentioned that: Assume topological group $G$ has a fundamental system of neighborhoods consisting of subgroups as: $G= G_0 \supseteq G_1 \supseteq\cdots\supseteq ...
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21 views

Topological Semi conjugacy between Henon map and Logistic Map

I am currently teaching myself dynamical systems and have come across a problem I am not quite able to figure out. More specifically, I am unable to find a conjugator function to establish a semi ...
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76 views

Precisely what is meant by “$\pi_1(M)$ is torsion”?

I am reading a paper where one of the conditions for a Theorem to hold is "the group $\pi_1(M)$ is torsion", where here $M$ is a compact differentiable manifold. What is meant by the first homotopy ...
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28 views

Stone-Cech compactification of a compact space and the multiplier algebra of $C_0(X)$

What is the Stone-Cech compactification $\beta X$ of a compact space $X$, is is $X$ itself? Or does it depend on the definition of compactification, whether the embedding $i:X\to \beta X$ is assumed ...
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33 views

Separability in Algebra and Topology

I am wondering about the use of the word separablity in two different areas of mathematics, namely algebra and topology. In topology, we call a topological space $(X,\mathscr{T})$ if it contains a ...
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41 views

Families of bounded (or closed) subsets of $\mathbb{R}^1$ with empty intersection, but having the finite intersection property

2.36 Theorem. If $\{K_{\alpha}\}$ is a collection of compact subsets of a metric space $X$ such that the intersection of every finite subcollection of $\{K_{\alpha}\}$ is nonempty, then $\bigcap ...
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14 views

Is there an analog of of equilateral triangles of height = leg on a sphere for a tetrahedron and a 3-sphere?

I've run into that on a unit sphere, an equilateral triangle of leg length = pi/2 (pole to equator) will have a height of pi/2. Does this generalize upward that a tetrahedron can be mapped on the ...
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50 views

Sum of the reciprocals topology

Let's define this topology in $\Bbb N$ (here $\Bbb N$ begins at $1$): $$K\subset\Bbb N\text{ is closed }\iff K=\Bbb N\;\text{ or }\;\sum_{n\in K}\frac1n\text{ converges}$$ I have worked some on it. ...
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36 views

Does there exist a kuratowski set which is uncountable

A subset of a topological space is called the Kuratowski set if we can get 14 different sets by applying closures and complementation successively. I want to find a set which is uncountable and is a ...
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72 views

How to write a “set is open” or “set is closed” in a pure symbolic way with quantifiers? (FOL)

How to write a "set is open" or "set is closed" or "a set is open" in a pure symbolic way with quantifiers? And how to use pure symbol to prove "E is open iff its complement is closed"? and that ...
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28 views

Complex Projective Space as a Quotient of a Disc

I am reading Hatcher's book and I have a problem understading how the complex projective space $\mathbb CP^n$ can be realised as a quotient of $D^{2n}$ (page 7) Let me briefly outline his arguments ...
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66 views

Is the unit square a $2$-manifold in $\mathbb{R}^2$?

I'm using the following definition of a (smooth) manifold: It's from J.Munkres "Analysis on Manifolds". This is an exercise taken from this book: Is the unit square $[0,1]\times [0,1]$ a ...
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37 views

Cartesian product between manifolds

I was given the following exercise: Show that if $M$ is a $k$-manifold without boundary in $\mathbb{R}^m$, and if $N$ is an $l$-manifold in $\mathbb{R}^n$, then $M \times N$ is a $k+l$ manifold in ...
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59 views

Continuous function $0$ on one closed set and $1$ on the other

Looking for a better approach of the following question if possible. Question: Let $A$ and $B$ be disjoint nonempty closed sets in a metric spaces $X$, and define ...
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34 views

homeomorphism as a result of other homeomorphisms

If $$B = \bigcup_{R>0} B_R$$ and all the identities $$\operatorname{id}_R : (B_R,d_1) \rightarrow (B_R,d_2)$$ for $R>0$ are homeomorphisms, then is $$ \operatorname{id} : (B,d_1) \rightarrow ...
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30 views

The set of limit points is closed

Let $E'$ be the set of all limit points of a set $E$. Prove that $E'$ is closed. Let $z$ be a limit point of $E'$. Then for any $\varepsilon>0$ deleted neighborhood with radius $\varepsilon/2$ has ...
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27 views

order topology and subspace topology

let $X=(-\infty, -1) \cup [0,\infty)$, subspace of $\mathbb{R}$. Then is it different from the order topology? Say $(-1/2,1) \cap X =[0,1)$ is open in $X$, but not open in the order topology??
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21 views

Show: if $x \in A'$, then $x \in (A \setminus \{x\})'$ in Hausdorff space; dash denotes limit points

EDIT: It was pointed out I managed to misstate the question. Sorry, and thanks to Daniel Fischer for pointing it out. RTP: Letting $B'$ be the set of limit points of $B$ for any set $B$, show that ...
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40 views

Homotopy equivalence of a 2-torus with 2 circles filled by disks to a 2-sphere.

How to show the homotopy equivalence of the union of the standard 2-torus with two disks, one spanning a latitudinal circle and other spanning a longitudinal circle of the torus, to a 2-sphere? I ...
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34 views

Showing a set of box topologies is homeomorphic

Let $(X,\tau_X) $ and $(Y, \tau_Y) $ be topological spaces. Consider $(X \times Y,\tau_{X \times Y}) $ and $(Y \times X, \tau_{Y \times X})$ where $\tau_{X \times Y}$ and $\tau_{Y \times X}$ are the ...
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22 views

Examples of space which is not totally bounded

I know some examples of spaces which are not totally bounded. For example, the real space $R$ with discrete metric is bounded but not totally bounded. I understand its not totally bounded because the ...
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17 views

Limits and Convergence of sequences in the form of $(k, k^2, 1/k)$

I'm dealing with proving the convergence and limits of sequences that are defined by multiple points, such as $$ \left(k, k^2, \frac{1}{k}\right) $$ and I'm not sure how to go about doing it. I'm ...
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37 views

Prove that the property of being isolated is a topological property

Let $f:X \to Y$ be a homeomorphism and $A \subset X$ such that $A \cap A'= \emptyset$ prove that $f[A] \cap (f[A])'= \emptyset$ where $A'$ is the derived set of $A$ and $(f[A])'$ is the derived set ...
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34 views

Construct a SPD kernel using a (true) distance function

Let $\mathcal{X}\equiv\Bbb{R}^n\times\Bbb{S}_{++}^n$ be a non-empty set of pairs $(\mathbf{x},\Sigma_x)$, where $\mathbf{x}\in\Bbb{R}^n$, $\Sigma_x\in\Bbb{S}_{++}^n$, where $\Bbb{S}_{++}^n$ denotes ...
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20 views

Can this be a way to prove that the set of all limit points is closed

Let $x\in A'$ where $A'$ is the set of all limit points of $A$ . Let $x$ be a limit point of A'. This implies every neighborhood $x$ contains some p such that $p\neq x$ and $p\in A'$. This implies ...
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33 views

Continuos, surjective map $\pi$ is a quotient map $\iff$ $\pi$ sends saturated open to open or saturated closed to closed

Problem is: Continuos, surjective map $\pi$ is a quotient map $\iff$ $\pi$ sends saturated open to open or saturated closed to closed. ($U$ is saturated $\iff$ $\exists V \in Y$ s.t. $U = \pi^{-1} ...
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43 views

Property of Cantor set

How to prove that no segment of the form $\left(\dfrac{3k+1}{3^m}, \dfrac{3k+2}{3^m}\right)$ where $k$ and $m$ are positive integers, has a point in commoin with Cantor set. How to prove this ...
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43 views

Are there any open topological spaces other than R3 with overall zero curvature (and asymptotic to R3 towards infinity)?

What I mean by this is as follows: Take an infinite flat manifold $\mathbb{R}^3$ with zero curvature. Then subtract out a knotted torus or linked tori. And sew them back in using Dehn surgery. (In ...
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35 views

Are points $x\in (-1,1)$ only interior points or also limit points in $\mathbb{R}$ with the usual topology?

Here on page 4 one writes: "A point $x$ is called a limit point of U if $(B_r(x)\setminus \{x\})\cap U\neq \emptyset $ for every ball around $x$." (The definition of a ball around $x$ and of an ...
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28 views

Homotopy and equivalent norms in a real Banach space

Let $X$ to be a real Banach space, endowed with a norm $\|\cdot\|$ and I show the following computations, after I ask to search understanding of these, because today I have gaps in functional analysis ...
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14 views

How to map pochhammer contour over normal Riemann surfaces?

Consider the simplest non-trival Pochhammer integral for the Beta function: $$ \int_P z^{1/2}(1-z)^{1/3}dz$$ The underlying algebraic function $w$ has genus $1$ so it's normal Riemann surface is a ...
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24 views

Continuous dense restriction makes the whole map an continusou mapping?

Let $X$ and $Y$ be two topological spaces and $f:X \rightarrow Y$ a map. Let $U$ be an open and dense subset of $X$ such that the restriction of $f$, $f_{|U}:U \rightarrow Y$ is continuous. Is then ...