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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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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53 views
+50

Creating a Topological group from modulo multiplication Group.

If I were to create a Topology out of the Modulo 3 Multiplication group $\mathbb{Z}_3$, what elements would it consist of and why? So $\mathbb{Z}_3 = \{0,1,2\}$ as a group over modulo 3. What are the ...
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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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50 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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19 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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49 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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30 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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31 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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16 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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12 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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23 views

How to extend a continuous function from domain open in $H^k$ to a domain open in $R^k$?

Assume $f:U\mapsto V$ is continuous and of class $C^r$, and $U$ open in $H^k$ but not $R^k$. How do I extend $f$ to $g:U'\mapsto V$, $g$ is also continuous and of class $C^r$, $U'$ is open in $R^k$, ...
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59 views

Connected metrizable spaces with more than one point

Let $X$ be a connected metrizable space with more than one point. Must $X$ contain a non-trivial path?
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44 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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31 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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41 views

If X is a space in the order topology with lub. If A is closed, is A compact?

In J R Munkres section 27, there is a theorem that states that every closed interval(note not ray) in the order topology where $X$ is a set with lub property is compact. I'm wondering if $X$ is a ...
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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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19 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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23 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!
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53 views

Finite covering space with compact spce.

Prove that if $p: \ Y \rightarrow X$ is finite covering, then if $Y$ is compact so it is X. Can someone check my attempt? :) Let $\mathcal{U}$ be any open cover of $X$. For every $x \in X$ let us ...
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26 views

every path in $X$ is homotopic with endpoints fixed to a path passing through $b$

$X$ is path connected and b$\in$X, show every path in $X$ is homotopic with endpoints fixed to a path passing through $b$ This is the hint in the book: Let $\gamma$ be a path from $x$ to $y$. If ...
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20 views

an infinite discrete subspace

**Each infinite subspace of a KC space contain an infinite discrete subspace.** Proof: Let $ (X,\tau)$ be aKC space, and $A ‎‎\subseteq‎ X$ is infinite. since $A$ does not have the cofine topolog , ...
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34 views

Show that the following Statements is true?

Let $\tau $ be the topology on $\mathbb R$ for which the interval $[a,b)$ form a base.Let $\sigma$ be a topplogy on $\mathbb R $ such that $\tau \subseteq \sigma$. Then If the map $ x \mapsto -x$ ...
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Confused about boundary points

Please pretend that these lines are zoomed-in parts of a disc in $\mathbb{R}^2$. The first picture on the left is a point immediately adjacent, or tangent to, the disc. The second picture on the ...
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46 views

Grothendieck topology

Hello : Here is a small parapgraph that i try to understand : If $ M $ is a smooth manifold, then we can recover the underlying set of $ M $ by considering the set $ \mathrm{Hom} ( \{ \star \} , ...
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23 views

Closure of a linear subspace of $C([a,b])$

Given the space $C([a,b])$ (the collection of all real-valued, continuous (with respect to the metric $d(x,y)=|x−y|)$ functions defined on the interval $[a,b]⊆R$), along with the uniform norm and the ...
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97 views

Measurable function implies equivalent to an exponential function.

This is a follow up to this question. In that question, I answered that an exponential function can be uniquely determined by three properties: a functional equation, a weak continuity assumption, and ...
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35 views

Removing a line boundary form a half disk

Let's cut a disk in two halves. Take one of the two halves. Its boundary is made of two pieces: one is the half circle and the other is "the line boundary" along the cut. I was told that the half ...
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53 views

$E$ is closed $\leftrightarrow E^{c}$ is open

I'm having difficulty following this proof provided in Principles of Math. Analysis by Rudin. Pf First suppose $E^{c}$ is closed. Choose $x \in E$. Then $x \notin E^{c}$, and $x$ is not a limit ...
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28 views

Some basic Topological proof help please.

Basically im really bad at proofs and i havent done math in almost a year and decided id like to learn topology on my own... just want someone to be really critical on my solutions please also i would ...
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21 views

Stone-Čech compactification not by ultrafilters only.

I am familiar with Stone-Čech compactification using ultrafilters. But, I, somehow can't understand the construction by commutative diagram, and certainly can not see the connection between the two ...
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32 views

Fibres, rammification points and continuity

Trying to understand the relation between the following: given a function $f:\mathbb{R} \to \mathbb{R}$ with all fibres either empty or of size 3, what can be said about the continuity of such a ...
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29 views

Covering dimension of a compact metric space

I would like to see the proof of the following fact (references appreciated). A compact metric space $X$ has covering dimension $\leqslant n$ if and only if there is a continuous surjection $\pi ...
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45 views

Complex analysis winding number

We have that $f:\mathbb S^1 \rightarrow \mathbb S^1 $ and $f(z)=f(1)\widehat{\phi}(z)$ with $\widehat{\phi}(\exp{2\pi it}) = \exp(2 \pi i \phi(t)),$ where $\phi:I \rightarrow \mathbb{R} $ is a ...
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42 views

Showing the bijection to prove that two sets are equivalents

I need to prove that two sets $X$ and $Y$ are equivalent by showing the bijection between them, where $X = \{x: Ax = c\}$ and $Y = \{y: Ay\le b,-Ay\le -b\}$. $A$ is a matrix, $x$,$y$,$b$ and $c$ are ...
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34 views

Show that $(1,1,1,…)$ is a limit point of a set $A$.

Let $X_j$ be $\{0,1\}$, the 2 point set, with the discrete topology for $j = 1,2,…$. Let $X$ be the countable product of the $X_j$'s with the product topology. Let A be the set which consists of ...