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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12 views

Presheaf of real valued functions

Seen as how a Presheaf of real valued functions on a topological space X associates a function f:U→ℝ to each open set U, what function maps the empty set to ℝ since the empty set is by definition an ...
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30 views

Prove that a defined function g is continuous for a certain point

Consider the set of rational numbers $\mathbb{Q}$ equipped with the subspace topology inherited from $\mathbb{R}$. Let $c \in \mathbb{R}$. Define the function $g_{c}: \mathbb{Q} \to \mathbb{Q}$ via: ...
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15 views

Topological conditions on compact Hausdorff $X$ under which the unital C* algebra C(X) is separable

If $X$ is a compact Hausdorff topological space, under which extra assumptions do we get that the unital $C^*$ algebra $C(X)$ is separable?
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41 views

Prove that $\mathbb{R} \times S^1$ is homeomorphic to $\mathbb{R^2} \setminus \{(0,0)\}$

I need to prove that $\mathbb{R} \times S^1$ is homeomorphic to $\mathbb{R^2} \setminus \{(0,0)\}$. I define the map $h:\mathbb{R} \times S^1 \to \mathbb{R^2} \setminus \{(0,0)\}$ by ...
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27 views

At most one connected component of unbounded portion of entire function.

Suppose $f$ is an entire complex analytic function and $R$ a positive real number. Define the set $E:= \{z\in\mathbb{C};|f(z)| < R\}$ to be the set of $z$ whose image is bounded by $R$. I want to ...
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25 views

Family of Morse functions made constant

I'm looking for a proof of the following theorem: Let $f_t$ be a family of real-valued Morse functions defined on a smooth compact manifold $M$, and where $t$ is in $[0,1]$ (So for all value of $t$, ...
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30 views

The set of rational numbers, each point is point accumulation

Please let us help someone by telling you a precise formulation is below, and then someone please tell me solution that has since become like that with a few days my friend we debates, here my ...
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19 views

Question about Boundary points of the sets in metric space

Let A be a metric spaces. Prove the following properties: The boundary of $A$ equals $A'-A$ The boundary of $A$ is the closed set. $A$ is closed if and only if it contains its boundary. Where ...
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38 views

Injective and continuous function that is an embedding

Consider $n,d\in \mathbb N$ and $N= {n+d\choose d}-1$, then the well known $d$-uple embedding: $$\rho_d: \mathbb P^n(\mathbb C)\longrightarrow\mathbb P^N(\mathbb C)$$ is a continuous (respect to ...
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45 views

Proof that a correspondence is upper hemicontinuous if and only if it's graph is closed

I'm working through a textbook (General Equilibrium Theory) where proofing the following theorem is left as an exercise to the student - unfortunately I dont know how. Theorem 23.1: (A ...
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65 views

Canonical topology on standard groups?

I just wanted to know whether there is any standard topology on groups like $\mathbb{Z}/n\mathbb{Z}$ or $\mathbb{Z}$ ? - The only one that I could imagine, especially for finite groups is the discrete ...
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15 views

Are (certain) metric-preserving vector bundle maps proper?

Given two real vector bundles $p\colon U \to X$ and $q\colon V \to Y$ with a metric and a vector bundle map $f\colon U \to V$ preserving this metric (i.e. it's fiberwise an orthogonal map). Can we ...
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22 views

Definition of a Paracompact space

I have a question about the definition of a paracompact space. We said that a space $X$ is paracompact iff $X$ is $T_2$ and if any open covering of $X$ has a finer locally-finite covering. I don't get ...
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88 views

If $X$ is compact and $f:X \rightarrow Y$ is a dense continuous injection, then $f$ is a homeomorphism

I found this: Let $X$ be a compact space and $f:X \rightarrow Y$ a continuous injection. Let $f(X)$ be dense in $Y$. Prove that $f$ is a homeomorphism. So, my question is: is it possible to prove ...
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38 views

Link complement simply-connected if codimension $\geq 3$

In Rolfsen, page 50 says that "an easy general position argument shows that a PL link $L^k$ in $S^n$ has simply-connected complement if $n - k > 3$," where $L^k$ is a $k$-dimensional link in $S^n$. ...
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23 views

Can anyone give the equation of the inverse of radial projection from a tetrahedron to sphere?

$(x,y,z) \mapsto \bigg(\frac{x}{\sqrt{x^2+y^2+z^2}},\frac{y}{\sqrt{x^2+y^2+z^2}},\frac{z}{\sqrt{x^2+y^2+z^2}} \bigg)$ This is the equation of the radial projection. I need the inverse of this ...
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18 views

Mapping open on open dense subset => Mapping is open on whole space?

Let $X,Y$ be topological spaces, and let $f\colon X \to Y$ be a continuous function. Further suppose that there exist an open and dense subset $U$ of $X$, such that $f\vert_{U} \colon U \to Y$ is an ...
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60 views

Zero set of finitely many polynomials.

Somebody asked a question earlier regarding this proof but I'm confused about a different part. I understand everything but the line "As the zero set of finitely many polynomials, $R$ is a closed ...
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68 views

Topology problem: Proving that sections are open

I have been trying to learn some basics of topology on my own, I have learnt the basic definitions. I have not been able to understand the proof provided in the text. Could anyone provide a clearer ...
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11 views

Definition of Non-characteristic Manifold

What is the definition of non-characteristic manifold in the following context. "Since $ f (x, y)=0 $ is assumed to be a non-characteristic singularity manifold, we have $ f_{x}\neq 0 $." Thanks, ...
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42 views

Finding a homeomorphism between these two balls

Let $u_1,u_2,u_3 \in \Bbb C$ be the cubic roots of unity. Define two norms on $\mathbb{C}^2$, $$\Vert (x,y) \Vert_1 = \sqrt{\vert x \vert^2 +\vert y \vert^2} \ \text{and} \ \Vert (x,y) \Vert_2 = ...
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37 views

Negation - Cauchy sequence

Suppose that $(x_i)_{i \geq 1}$ is not a Cauchy sequence of real numbers. How to prove that there exist $\varepsilon >0 $ and an increasing sequence $(i_n)$ of indices such that $$ ...
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47 views

Can a space that is not locally finite be paracompact?

Can a space that is not locally finite be paracompact? I am not convinced of this fact, but I am told that examples exist.
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46 views

Proper maps and Hausdorff Space

Now, I need help with this exercise: Let $(X,\tau)$ and $(Y,\sigma)$ two topological spaces. Let $f:X\to Y$ a continous map such that the preimage of a compact subset in $Y$ is compact in $X$. Show ...
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51 views

Interior of a Dirichlet domain in a Riemannian manifold

Let $X\neq\varnothing$ be a complete connected Riemannian manifold. Suppose $G$ is a group of isometries of $X$, acting properly discontinuously on $X$. We assume there is a point $x_0\in X$ such that ...
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30 views

Separable metrics

Is the following true?: Let $d_1$ and $d_2$ two separable metrics in space $X$. Then $d=\max(d_1,d_2)$ is a separable metric on $X$. Thanks!
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18 views

How to generate a Poincare section for discrete particle trajectory?

I'm a novice when it comes to generating Poincare sections, and I can't seem to get it right. I have a particle moving in a 3D periodic field, and I wish to generate a Poincare section of its ...
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20 views

A Lindelof non-scattered space $X$ which is not an extention of $\mathbb R$

Is anyone familier with an example for a Lindelof non-scattered topological space $X$ which is not an extention of $\mathbb R$ (with Euclidean topology). I am looking for an example which is not a ...
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16 views

Proof verification related to the discrete metric

Can someone please verify my proof? Let $X_1$ be a set and let $d_1$ be the discrete metric on $X_1$. (a) Prove that every subset of $(X_1, d_1)$ is open. (b) Prove that if $(X_2, d_2)$ ...
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19 views

Showing that bd$A$=$\{\vec v\in\mathbb{R}^n| d(\vec u,\vec v)<r\}$

Problem: And $\beta_r(\vec u)\equiv \{\vec x\in\mathbb{R}^n| dist(\vec u,\vec x)<r\}$. I got the first part showint that Int $A$=$A$. Now I want to show that bd$A$=$\{\vec v\in\mathbb{R}^n| ...
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14 views

Infimum simple function and stepfunction

If $A_n$ is a simple function and $B_n$ is a Stepfunction, then the infimum ($A_n\wedge B_n$) is a stepfunction. Why is this true?
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50 views

Compact-Open topology on C(X).

Let $C(X)$ denote the set of all real-valued continuous functions on a Tychonoff space X. And for a subset $A$ of $X$ and subset $V$ of $\mathbb{R}$, let $[A, V] = \{f \in C(X): f(A) \subseteq V\}$. ...
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25 views

Problem concerning proof that the set $A=\{x_n\},$ where $x_n=(1/n) \times n$ is a closed subset of $X=[0, 1] \times \{n\}, n \in \mathbb{Z_+}$

I understand that to show that the set is closed, I need to show it has no limit points. How do I do that?
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31 views

$\omega_1$ contains an initial segment as a copy of $\omega + 1$

Here, $\omega + 1$ is defined to be the union of Natural number and a number larger than all natural numbers. I know that the number larger than all natural number in this set works as a limit point ...
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27 views

Find close points by grouping points in n-dimensional space

I know I already asked a similar question over here: Finding the closest point in a set to another point in n-dimensional space: efficiently But now my question is different. I have many points ...
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13 views

What is the (pre)topology hinted by the Daniell integral

Daniell takes a vector lattice $H$ of a set $\mathbb{R}^X$ which he calls the set of elementary functions. For him, an elementary integral $I$ is a nonnegative functional on $H$ which verifies : if ...
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16 views

Name for point in an infinite product of intervals bounded away from the boundary

Consider $[0,1]^{\omega}$ (the product of a countable amount of $[0,1]$ intervals). I am interested in points $x=(x_1,x_2,\ldots)\in [0,1]^{\omega}$ that are bounded away from the boundary, i.e. ...
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23 views

Need help with the proof of the KKM-lemma.

I have been working on the proof of the KKM-lemma, which states Let $\lbrace A_0,A_1,...,A_n \rbrace$ be a closed covering of an $n$-simplex $\sigma=[x_0,...,x_n]$ such that for each face ...
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32 views

Order of refinement of an open covering of $X$, a metric space

If every finite open covering of a metric space $X$ has a refinement of order $\leqslant n$, is it true that every open covering does too? We say that a covering has order $n$ if $n$ is the largest ...
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26 views

The number of non-degenerate proper subcontinua in a non-degenerate continuum

A continuum is any compact connected metric space. A continuum is non-degenerate if it is not a single point. My question is thus this: How many non-degenerate proper subcontinua must a ...
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17 views

Is there a term in Topology for a boundary point that belongs to a given set?

If M is a subset of a topological space and U is the interior of M, is there a standard name for the set M - U? In other words, is there a standard name for the set of boundary points of a given set ...
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14 views

Topological semi-conjugacy between two systems

I have a few questions regarding the clarification of the difference between semi-conjugacy and conjugacy. I am trying to figure out a explicit conjugator function to semi-conjugate (a-x^2, x) and ...
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45 views

one point compactification of upper half plane

"prove that the closed unit disc could be considered as the one point compactification of the upper half plane with the X-axis as the bounday" i believe in this statement by the imagination,but i ...
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32 views

Choosing a canonical fundamental domain

I have a set of equations that partitions a certain space into equivalent regions. For a given point $p$ contained in region $R_1$, there are equivalence relations giving its equivalent position in ...
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22 views

reverse-reverse of Michael selection theorem

Let $X\subseteq\mathbb R^d$ be a compact and $Y=\mathbb R^d.$ Let $\Gamma:X\twoheadrightarrow Y$ be a multi-valued map with closed values. Assume that $\Gamma$ admits a continuous (single-valued) ...
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28 views

Relational calculus. Natural join

How does natural join look in relational calculus? I think it looks like {A1, A2, A3, A4 | r1(A1, A2, A3) AND r2(A2, A3, A4)} Correct me please if I am wrong. ...
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20 views

A question about the Zariski space.

A Zariski space is a topological space with the property that every descending chain $F_1\supset F_2\supset F_3\dots$ of closed sets is eventually constant. Show that every Zariski space can be ...
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38 views

finite subset topology

Let $X$ be a set and $\tau=\left\{X\right\}\,\cup\,\{u_i\subset X|u_i \text{ is finite}\}$. Is $\tau$ is a topology on $X$? My effort to show this is as follows: 1) $X\in\tau$ by definition and ...
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24 views

The closure of semialgebraic sets is semialgebraic.

I want to prove that the closure of semialgebraic subsets of $\mathbb{R}^n$ with respect to the Euclidean topology is semialgebraic. I may use the Tarski–Seidenberg theorem. Please give me not the ...
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23 views

Closure A union A with accumulation points

Can somebody help me by proving that the closure of A is the union of A with the set of acculumation points of A? From the left to the right is easy, but the other way isn't clear for me. Thanks!