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

The Coproduct of two spaces is the same as the disjoint union and is homeomorphic to the union when the spaces are disjoint

Let $X$ and $Y$ be two topological spaces. Consider the set $$X \cup Y = \{ x\; |\; x \in X \text{ or } x \in Y\}$$ In all the following I suppose that $X$ and $X$ are disjoint. I want to ...
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4answers
47 views

What are some simple examples illustrating the definition of “cover”

In my class the word "cover" is used very informally such as this set covers another set (this is for a class in PDE not topology by the way). Can someone provide a trivial example of cover to get ...
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8answers
176 views

What is the mathematical distinction between closed and open sets?

If you wanted me to spell out the difference between closed and open sets, the best I could do is to draw you a circle one with dotted circumference the other with continuous circumference. Or I would ...
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2answers
26 views

A base generates an unique topology?

I was confused by this. Let $X$ be {$a,b,c$}, Let $\mathcal{B}$ be {{$a$},{$b$},{$c$}}. Let $ \mathcal{T}$ be {$X, \emptyset$, {$a$}, {$b$}, {$a,b$}}. Let $ \mathcal{T'}$ be {$X, \emptyset$, ...
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0answers
13 views

Disjoint Union of Completely Regular Spaces

I am trying a new approach to an already-solved problem, but I need help to see if I'm on point. Munkres Chapter 53, question 6 [abridged] asks, given a covering map $p: E \to B$: Show that "if $B$ ...
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2answers
59 views

Decomposing 2-sphere into two homeomorphic subspaces [on hold]

Can a 2-dimensional sphere be decomposed into two disjoint homeomorphic subspaces? If yes, can these subspaces be non-discrete / connected / have some other good properties?
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0answers
10 views

Construction of a Radon measure from a certain family of compact subsets

Let $X$ be a locally compact Hausdroff space. Let $\Gamma$ be a family of compact subsets of $X$ with the following properties. 1) $\phi \in \Gamma$. 2) $K\cup L \in \Gamma$ whenever $K \in ...
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1answer
37 views

simplicial homology definition

I am revising for my algebraic topology exam based on Hatchers 2nd chapter of algebraic topology and I have this question regarding the definition of simplicial homology on pages 104-105: Hatcher ...
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1answer
21 views

Limit vs interior definition of continuity

Suppose I have two topological spaces $X$ and $Y$ whose topologies are defined by interior operators $\text{int}_X$ and $\text{int}_Y$ respectively, as well as a function $f$ with domain $I$ (for ...
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1answer
21 views

Boundary preserving map

Let $K\subseteq\mathbb{R}^2$ be a compact set. Is it true that for a continuous map $p:K\to\mathbb{R}^2$ we have: $p(\partial K)=\partial p(K)$? Are there any generalizations? P.S. Note that ...
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0answers
24 views

Pro-completion of finite algebras as Stone algebras

Recall that a profinite algebra (e.g. group, monoid, or whatsoever) is a cofiltered/inverse limit of finite algebra. In Johnstone's Stone space, he showed that finite discrete algebras are finitely ...
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1answer
26 views

Show that if $(X,d)$ is compact then, every open covering of $X$ has a Lebesgue number.

Let $(U_i)_{i \in I}$ be an open cover of a metric space $(X,d)$, a number $\epsilon >0$ is called a Lebesgue number of $(U_i)_{i \in I}$ if for all $x \in X$ exist $j \in I$ such that ...
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2answers
42 views

An infinite dimensional normed linear space is the union of two disjoint convex sets

Let $X$ be an infinite dimensional normed linear space. I want to show that there exist two disjoint convex sets $C_1$ and $C_2$ such that $X=C_1\cup C_2$ and both $C_1$ and $C_2$ are dense in $X$. I ...
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1answer
27 views

Continuous function between two topological spaces: an ELEMENTARY property. [duplicate]

I'm reading the first chapter of a book on general topology. It has a lot of small, simple exercises on almost all pages and I try to do them all to fully understand the subject. This one I did not ...
2
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0answers
25 views

If a set is Hausdorff relative to one topology, can it be compact relative to a strictly finer topology?

Let $\tau_1$ and $\tau_2$ be two topologies on a non-empty set $X$ such that $(X, \tau_1)$ is Hausdorff and $\tau_1 \subsetneq \tau_2$. Can $(X, \tau_2)$ be compact? My effort: Suppose that ...
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2answers
61 views

Is this condition on continuity extraneous or troublesome?

I was trying to motivate the use of open sets for defining continuity (as in topology or metric spaces). I came to formulate the following definition of continuity for a function $f: X \rightarrow ...
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0answers
19 views

Metric spaces and compactness [on hold]

Let $X$ be a metric space. If for all compact $K$, the set $K\cap F $ is closed, then $F$ is closed.
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3answers
31 views

Statment concerning open sets and closures

I found the following line in a proof (from a good book) concerning locally compact spaces: Since $A$ and $B$ are both open and $A \cap B = \varnothing$, it follows that $\bar{A} \cap B = ...
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0answers
17 views

Compactness of a convex collection

Given $\epsilon\in(0,1)$, suppose we have collection $\mathscr{C}(\epsilon)$ of multilinear polynomials in $\Bbb R[x_1,\dots,x_n]$ that on $\{0,1\}^n$ is in range $(-\epsilon,\epsilon)$ on $S_0$ while ...
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0answers
28 views

How is this convex set compact as well?

Given $\epsilon\in(0,1)$, supposing we have a collection $\mathscr{C}(\epsilon)$ of polynomials in $\Bbb R[x_1,\dots,x_n]$ that on $\{0,1\}^n$ takes on value $0$ on $S_0$ while being in range ...
3
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1answer
88 views

Maximum $C$ such that every shape in $\Bbb R^2$ with area $<C$ can be placed to avoid $\Bbb Z^2$

For $C=1$, it has been proved here that every shape in the plane having area less than $1$ can be translated and rotated so that it does not touch any element of $\mathbb Z^2$. (In fact, for $C=1$, ...
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4answers
267 views

Little confusion about connectedness

Consider $X=\{(x,\sin(1/x)):0<x<1\}$. Then clearly $X$ is connected , as it is a continuous image of the connected set $(0,1)$. So, $\overline X$ is also connected , as closure of connected set ...
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1answer
114 views

Is it possible to develop differential geometry without points?

I read about pointless topology and locale theory, and become curious about this topic. For example, there is the concept "differential manifold" corresponds to "topological manifold". As this, are ...
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1answer
21 views

How to make a topology out of $N$ that involves convergent / divergent sets.

Let $N$ be the naturals $1, 2, \dots$ Call a subset $A$ of $N$ convergent if the reciprocal sum $\sum_{a \in A} \frac{1}{a}$ converges. Similarly call as set divergent if the sum diverges. Notice ...
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2answers
30 views

Three questions from σ-compact spaces and topological groups

every locally compact subgroup of a Hausdorff group is closed. A Hausdorff and $σ-$compact space X is a Baire space if and only if the set of points at which is $X$ is locally compact is dense in ...
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1answer
28 views

A locally compact Hausdorff space which is not a metric space

I need three simple examples. A locally compact Hausdorff space which is not a metric space. A locally compact topological group which is not Hausdorff. (my definition of topological group does not ...
2
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1answer
33 views

Haar measure on locally sigma-compact metric groups

Haar measure on locally sigma-compact metric groups $G$ is a metric group, if $G$ is a topological group meanwhile $G$ is a metric space(compatible with topology). We know that there exist a Haar ...
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0answers
36 views

On the properties of an interesting set on the real line…

Let $K$ be the set of all real numbers of the decimal form $$ 0.\;e_1\;\underbrace{0}_{1!\text{ times}}\;e_2\;\underbrace{00}_{2!\text{ times}}\;e_3\;\underbrace{000000}_{3!\text{ ...
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2answers
28 views

Find the limit points and exterior points of the following

Let $X=\mathbb R$, with the usual metric on $\mathbb R$ and $A=((0,1)\cap \mathbb Q)\cup$ {$2,3$}. Find the limit points of $A$, exterior points of $A$, $A^o$, $\overline A$ and $\partial A$. Can ...
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0answers
36 views

Show that subspace metric induces subspace topology [on hold]

Let $(X,d)$ be a metric space, let $\tau$ be the topology on $X$ induced by $d$ and $A \subset X$. Define $d_A: A \times A \to \mathbb R$ as $d_A(a,b)=d(a,b) \forall a,b \in A$ . Show that $d_A$ ...
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1answer
19 views

Existence of a basis for a topological space

I came across the following lemma: Let $X$ be a topological space. Suppose that $\mathcal C$ is a collection of open sets of $X$ such that for each open set $U$ of $X$ and each $x$ in $U$, there ...
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1answer
31 views

Show that $\tau_A$ is a topology on $A$

Let $(X,\tau)$ be a topological space and $A \subset X$. Let $\tau_A$={$A \cap U: U \in \tau$}. Show that $\tau_A$ is a topology on $A$. I know that I need to prove three properties to prove ...
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0answers
22 views

If a bounded sequence is equicontinuous, it has a uniformly convergent subsequence

I am currently having some difficulty with problem 2.7.8 in Introduction to Topology by Theodore Gamelin and Robert Greene. The problem goes as follows A family F of real-valued functions on a ...
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0answers
20 views

Prob. 1, Sec. 27 in Munkres' TOPOLOGY, 2nd ed: How to show that the compactness of every closed interval implies the least upper bound property?

Let $X$ be an ordered set in which every closed interval is compact. Then $X$ has the least upper bound property. How to prove this? My effort: Let $A$ be a non-empty subset of $X$ such that $A$ is ...
2
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1answer
32 views

For an orientable surface of genus $n$ what is the maximal number of noncontractible loops that can be drawn on that surface?

This is related to a homework question in a condensed matter course. For each noncontractible loop which can be drawn on a lattice wrapped onto a surface of genus $n$ we can define two operators. Each ...
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0answers
29 views

Transversality of graphs of functions

Consider the $C^1$ function $f: [0,1] \to \mathbb{R}$. I understand that a curve in the plane that intersects the graph of $f$ non-transversally would be tangent to it at a point of intersection. I ...
2
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2answers
50 views

Bounded complete metric space is compact?

This question may seem trivial, but in topology we were taught that in a complete metric space, a subset of that space was compact if and only if it is closed and bounded. Moreover, we are told that ...
2
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2answers
57 views

In topology, any relationship on boundary like $Bd(A\cap B)$ and $Bd(A) \cap Bd(B)$?

Given a topological space $(X,\mu)$, let $Bd(S)$ denote the boundary of subset $S\subseteq X$. Is there any relationship between $Bd(A\cap B)$ and $Bd(A)\cap Bd(B)$ for arbitrary subsets ...
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1answer
31 views

Question regarding subspace and order topology

In Munkres Book, Pg 90, I came across this question in Example 2: Let $Y$ be the subset $[0,1)\cup\{2\}$ of $\mathbb{R}$. In the subspace topology on $Y$, the one-point set $\{2\}$ is open as it is ...
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1answer
21 views

Proof cube & trapezium a Compact Space & $E^n$ & $I^n, I^{\infty}$ are connected space ??? [on hold]

I need a serious help here please! Question 1: Prove that $E^n$ & $I^n$, $I^{\infty}$ are connected spaces. After a lot of search I found some two theorems in James Dugundji book. But I still ...
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0answers
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graphical topography [on hold]

an alloy containing 8cm3 of copper and 7cm3 of tin has a mass of 122.3g. A second alloy containing 9cm3 of copper and 7cm3 of tin has a mass of 131.2g. Using graphical methods find the densities of ...
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0answers
19 views

On the covering dimension of an image under a continuous function

I'm trying to solve the following exercise: Let $X$ be a compact Hausdorff space and let $U_1,...,U_n$ be a cover of $X$ of order $m$. Let $z_1,...,z_n\in\mathbb{R}^N$ for some $N$ be in general ...
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1answer
40 views

Show that S is closed but not compact

Show that $S$={$(x,y,z)\in \mathbb R^3: x^3+y^4-z^2=1$} is closed but not compact where $\mathbb R^3$ is the usual topology. Can anyone explain how to go about answering this? I have to show that ...
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1answer
19 views

Is the Sorgenfrey Line second countable? [duplicate]

The Sorgenfrey topology on $\mathbb{R}$ is the topology whose basic open sets are of the form $[a,b)$ where $a < b \in \mathbb{R}$. Does it have a countable base? (I suspect not.) Certainly it is ...
2
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1answer
24 views

Partial mapping, how does the inverse look like?

Let $X_1,X_2,Y$ be topological spaces. Let $f:X_1\times X_2 \to Y$ be continous at $a=(a_1,a_2)$. Show that the partial mappings $f_1:X_1\to Y; x\mapsto f_1(x) = f(x,a_2)$ is continous at $a_1$ and ...
2
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1answer
21 views

Pointwise Convergence: No Diagonal Subsequence Exists?

Can anyone find a sequence of arbitrary functions $f_n : \mathbb{R} \to \mathbb{R}$ that converge pointwise to an arbitrary function $f : \mathbb{R} \to \mathbb{R}$, such that for each $n$, there is a ...
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1answer
34 views

Fundamental groups in path-connected space

I'm studying Fundamental groups and today I saw the follow theorem: Theorem: Let be $X$ a topological space path-connected and $x,y\in X$. Then, the application $\psi:\pi_1(X,x)\to \pi_1(X,y)$ is a ...
4
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1answer
91 views

If the action is free, is it necessarily a covering space action?

Suppose a group $G$ acts simplicially on a $\Delta$-complex $X$, where "simplicially" means that each element of $G$ takes each simplex of $X$ onto another simplex by a linear homeomorphism. If the ...
2
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1answer
25 views

$|\beta(s)-\alpha(x)|<d$ implies that $\beta$ is homotopic to $\alpha$.

Let $D$ be an open subset in $\mathbb{R}^n$. Let $\alpha$ be a path in $D$ from $x$ to $y$, and set $d=\inf\{|\alpha(s)-w|:w\in \partial D, 0\le s\le 1\}$. Show that if $\beta$ is any path in $D$ ...
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2answers
25 views

A Hausdorff, Baire space must be σ -compact?

Must a Hausdorff Baire space be $σ-$compact? A Baire space is a topological space in which the union of every countable collection of closed sets with empty interior has empty interior. A ...