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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62 views

I want to self study systematically pure mathematics? Where do I start? [on hold]

I am an undergraduate student in Mechanical Engineering and I am highly interested in studying pure mathematics systematically.I have a fair amount of knowledge on real and complex analysis, ...
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1answer
20 views

Proof verification: Compact set has sup and inf

I was reading this post compact set always contains its supremum and infimum There was an answer reposted as follows: As $K$ is compact, we have that $K$ is bounded. So $\sup K$ and $\inf K$ ...
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1answer
19 views

Comparing different topological spaces regarding homeomorphisms and fundamental groups.

Which of the following topological spaces are homeomorphic? Which have the same fundamental group? a) The interval $(0,1)$ and $\mathbb{E}^1$ b) The torus $\mathbb{R}^2/\mathbb{Z}^2$ ...
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1answer
10 views

Effective Topological Transformation Groups and the Group of Homeomorphisms

I'm reading Steenrod's Topology of Fibre Bundles, and on pages 6 and 7, he defines a topological group $G$ and a topological transformation group of a topological space (which I understand to be a ...
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1answer
28 views

Show there are infinitely many distinct maximal solutions of $\frac{dx}{dt} = (3/2)x^{1/3}$ that pass through the point $(t_0,0)$

$$\frac{dx}{dt} = (3/2)x^{1/3}$$ Solve Show that given any point $(t_0,0)$ on the $t$-axis, there are infinitely many distinct maximal solutions that pass through the point. We are given: ...
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1answer
37 views

Why is the map $f(x)=e^{i2\pi x}$ from $[0, 1)$ to the unit circle continuous?

This seems to be a really silly question, I just couldn't think it straight. The definition of a continuous map: $f: X \to Y$ is continuous if for any open set $U$ in $Y$ , $f^{-1}(U)$ is open in ...
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1answer
20 views

Projection maps are open

I want to show $p_x: X\times\ Y \to X$ is an open map. Here's my proof: Let $W \subset\ X\times\ Y$ be open subset, then $W = \bigcup U_\alpha \times\ V_\beta$, for $U_\alpha, V_\beta$ are open ...
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20 views

shrinking a convex hull around a set of polygons

I'm trying to find (Or design) an algorithm that will let me, after I have a convex hull, progressively shrink the hull towards the polygon set via increasing some parameter. I.e., if we use the ...
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1answer
10 views

Suborderable space, orderable characterization proof doubt

In Orderability in the presence of local compactness, Valentin Gutev states and proves the following proposition: A suborderable space $X$ is orderable with respect to a linear order $\prec$ on it if ...
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4answers
428 views

A Compact Hausdorff Space with no Manifold Structure?

What is an example of a compact Hausdorff space that cannot be given the structure of a (i) differential manifold (ii) topological manifold?
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1answer
32 views

When do two subbases generate the same topology

Let $X$ be a set. If $\mathcal B_1$ and $\mathcal B_2$ are bases of subsets of $X$, it is well-known that $\mathcal B_1$ and $\mathcal B_2$ generate the same topology if and only if for any pair of ...
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1answer
47 views

a topological property of the product topology

Let $G$ be a non discrete Polish group. Let $K$ be a compact set of $G$, $C$ a closed set of $G^n$ and $B$ an open set of $G^n$. Suppose $K^n\cap C\subseteq B$. Prove that there is an open set of $G$, ...
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1answer
28 views

Closed subsets of $\mathbb{C}^*$ proper for multiplication

Let $S_1$ and $S_2$ be two proper closed subsets of $\mathbb{C}^*$. Let's denote by $\overline{S_1}$ and $\overline{S_2}$ their closure in $\mathbb{C}_{\infty}.$ (Alexandrov compactification) ...
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4answers
43 views

A sequence in a Hausdorff space and in a space that is not Hausdorff.

Let $X$ be a topological space and $\{x_n\}_{n=1}^{\infty}$ a sequence in $X$. Show that if $X$ is Hausdorff, $x_n \rightarrow x \:$, $x_n \rightarrow y \:$ implies $x=y$. Give an ...
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0answers
35 views

The closure of the closed ball is a closed [on hold]

In how many ways you can show that $\overline{\overline{B}(x,r)}=\overline{B}(x,r)$ where $\overline{B}(x,r)=\lbrace y \in \mathbb{R}^n : d_e(x,y) \leq r \rbrace$, and $d_e$ is the euclidean metric ? ...
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1answer
18 views

Sequential compactness of smooth functions

Suppose I have a sequence $u_n$ of smooth functions on the $N$-dimensional reals. If $\|D^{\alpha}u_n\|_{\infty} \leq C_{\alpha}$ for all multi-indices $\alpha$, then is it possible to deduce that ...
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1answer
59 views

are two metrics with same compact sets topologically equivalent?

are two metrics with same compact sets topologically equivalent ? I think if the cardinal of set is finite then we have one metric that is the discrete metric and every metric on this set is ...
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50 views

Is the sum of infinitely many open sets open?

Let $X$ be a locally convex space (or, in particular, a normed space). Let $(O_n)_{n=1}^\infty$ be an infinite sequence of non-empty open sets in $X$ such that the sum $\displaystyle\sum_{n=1}^\infty ...
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3answers
61 views

Show that two topological spaces are not homeomorphic.

Let $X = (-1,1)$ be considered with the Euclidean metric, and $Y = (0, \infty)$ be given the cofinite topology. Prove that $X$ and $Y$ are not homeomorphic. My current thoughts are that a ...
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3answers
62 views

Let $F : X → X$ be continuous. Prove that the set $\{x ∈ X : F(x) = x\}$ of fixed points of F is closed in X

Here X is a Hausdorff Space. I know that singleton sets, {x}, are closed in a Hausdorff space. Although Im not sure if thats how to use the Hausdorff property. Should I investigate $h=F(x)-x$? Can ...
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1answer
21 views

Show that $|d(m,n) -d(n,o) | \leq d(m,o)$ for a metric space

Problem Let $(M,d)$ be a metric space. Show that $$|d(m,n) - d(n,o)| \leq d(m,o) \ \forall m,n,o \in M$$ Since $(M,d)$ is a metric space I know it fufills the triangle inequality. So if I ...
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0answers
38 views

Is $(\omega \times \omega)^{\omega}\cong \omega \times \omega \times… \cong \omega^{\omega}$? Where “$\cong$” means homeomorphic.

I'm interested in the circumstances for when we can conclude that two ordinal spaces are homeomorphic by an examination of their written form. Specifically, I'm taking an ordinal space, say ...
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15 views

Special case of noetherian space

A topological space  is called Noetherian if it satisfies the descending chain condition for closed subsets. Now let $X $ be a topological space, and there exists a fix natural number $n $ such that ...
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22 views

Another question from Exercise 6d in section 50 in Munkres' textbook in Topology.

I have a question regarding exercise 6d in section 50 from Munkres' Topology textbook: Exercise 6c in section 50 Munkres' Topology textbook. Show that if $N=2m+1$, then $U_\epsilon(C)$ is dense ...
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20 views

Cantor-Bendixson rank of a first countable space

This question has been bothering me for quite a while, so let me ask it here. Is there a first-countable compact space $X$ with uncountable Cantor-Bendixson index? By a Cantor-Bendixson index I ...
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3answers
43 views

Countable connected spaces

I can not think of any countable connected subsets in $\mathbb{R}$ (with subspace topology).. Are there any such? Only countable subsets of $\mathbb{R}$ that i am familiar with is $\mathbb{Q}$ ...
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1answer
8 views

Does continuity in one variable and locally Lipschitz in another imply uniformity in the first?

I understand the definition of Lipschitz functions when talking of functions of single variables. However, I have trouble understanding it when it is a multivariable function. Suppose $ f(t,x):D ...
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1answer
39 views

composition of functions is continuous

Question is as follows : Let $X,Y,Z$ are metric Spaces Let $f:X\rightarrow Y$ be continuous map onto $Y$ and let $X$ be compact. Also $g:Y\rightarrow Z$ such that $g\circ f:X\rightarrow Z$ is ...
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1answer
37 views

Convergent Bounded Linear Maps

I'm not sure how to show that the composition of two convergent bounded linear maps converges to the composition of their limits. I've shown that the composition of bounded linear maps is a bounded ...
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1answer
25 views

topological invariance of being contained in a set of given dimension

Suppose $U$ is contained in $E^n$ ($n$-dimensional Euclidean space) and is homeomorphic to a set $V$ in $E^m$, where $m>n$. Is there a topological manifold in $E^m$ of dimension $n$ that contains ...
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2answers
43 views

Prove that the set $E = \{y ∈ Y : f(y) = g(y)\}$ (a.k.a. the coincidence set of $f$ and $g$) is closed in $Y$

The full question is: Let $X$ be a Hausdorff topological space. (i) Let $Y$ be a topological space and $f, g : Y → X$ be continuous functions. Prove that the set $$E = \{y ∈ Y : f(y) = g(y)\}$$ ...
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1answer
46 views

Fréchet derivative of $f(x) = x$

Im not sure how to find the Fréchet derivative of the function $f : \mathbb{X} \to \mathbb{X}$ given by $f(x) = x$, where $\mathbb{X}$ is a normed space. I'm not given the dimension of the normed ...
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86 views

Is every topological group the topological fundamental group of an space?

The fundamental group $\pi_{1}(X)$ of a path connected topological space $X$ is the image of $Hom(S^{1},X)$. So the fundamental group can be topologized with quotient topology where ...
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2answers
24 views

Showing the disjoint union topology is a topology

Let $A$ be a set and suppose that for all $\alpha \in A$, we have the topological space $X_\alpha$. Consider the set which is the disjoint union $$ X:=\coprod_{\alpha \in A} X_\alpha. $$ Let $\tau$ be ...
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1answer
22 views

An example of the set of distances of two points in two different closed sets having no infimum

On a problem set for my Analysis in Several Dimensions class (basically real analysis on multivariable functions), I encountered this question: Let $(X, d)$ be a metric space, let $C ⊂ X$ be a ...
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1answer
27 views

Is every compact metric space hereditarily separable?

Let $X$ be a compact metric space. I see why all open and closed subsets of $X$ are separable. But is every subset of $X$ necessarily separable? EDIT: Since $X$ is separable metric, it embeds into ...
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1answer
38 views

Does $X^{C_2} \simeq * \simeq X/{C_2}$ imply $X \simeq *$?

What the title says. Let $C_2$ be the cyclic group of order 2, and $X$ be a topological space with a $C_2$-action (acting continuously) such that both the quotient space $X/{C_2}$ and the subspace of ...
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2answers
41 views

If $X$ is a set and $\mathcal T$ is the discrete topology on $X$, is the following statement true

If $X$ is a set and $\mathcal T$ is the discrete topology on $X$, is the following statement true: $\{X\} \in \mathcal T$? I know that since $\mathcal T$ is a topology we know that $X ...
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1answer
44 views

Is one-point compactification of a space metrizable

Let $X$ be a locally compact Hausdorff space.Let $Y$ be the one-point compactification of $X$. Two questions are: Is it true that if $X$ has a countable basis then $Y$ is metrizable? Is it true ...
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0answers
32 views

Group theory: Intuition as to what a group is [duplicate]

In group theory the group is an algebraic structure consisting of a set which has elements associated with definite finitiary operations. Can an intuitive explanation be provided as to what this ...
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32 views

Why is the unit disc not a topological surface? [duplicate]

I am trying to prove that the unit disc $D^2$ is not a topological manifold. Clearly it is Hausdorff and second countable, so I think I should show that it is not locally Euclidean. The following is ...
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14 views

Ways to prove that two manifolds are $not$ ambient-isotopic to each other

I've just started learning basic topology and have just received an introduction to isotopy, so I apologize if this question appears trivial. What are some of the ways to prove that two manifolds are ...
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23 views

Intersections: Generator

Problem Given a set $\Omega$. Define the generator: $$\mathcal{A}\subseteq\mathcal{P}\Omega:\quad\delta\mathcal{A}:=\{A\cap A':A,A'\in\mathcal{A}\}$$ Then one obtains: ...
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1answer
55 views

What is boundary of $\mathbb{C}$? [on hold]

What is boundary of $\mathbb{C}$? or $\partial \mathbb{C}$?
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3answers
34 views

Show that these metrics induces the same topology on X

Let $X$ be the set of positive integers. Let $d_1$ be the usual metric space on $X$ and $d_2$ be the discrete metric on $X$. Define $d_3:X\times X \rightarrow R$ by ...
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10answers
3k views

Explain “homotopy” to me [on hold]

I have been struggling with general topology and now, algebraic topology is simply murder. Some people seem to get on alright, but I am not one of them unfortunately. Please, an answer I need is ...
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1answer
79 views

Exercise 6c in section 50 Munkres' Topology textbook.

The problem is as follows: Given $f: X \to \mathbb{R}^N$ and given compact subspace $C$ of $X$ ($X$ is locally compact Hausdorff space with a countable basis); let: $$U_\epsilon(C) = \{ f: ...
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3answers
53 views

Topological Continuous Functions and Non-Open Sets

Let us consider a function $\ \mathbf F $ defined from $\ \mathbf X $ to $\ \mathbf Y $ , where $\ \mathbf X $ and $\ \mathbf Y $ are topological spaces. Now by definition , a continuous function is ...
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34 views

Simply connectedness of spherical shell

Consider a spherical shell $U$ in $R^3$(the open region between two spheres). I want to show that any closed curve in $U$ can be shrunk into a single point without leaving $U$. This exercise appears ...
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38 views

What does a variable superscript above a set mean?

I'm not entirely sure I've worded this correctly. An example of what I mean is... $$U = \{0,1\}^n$$ What is the meaning of the superscript?