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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1answer
18 views

Decomposing 2-sphere into two homeomorphic subspaces

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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5 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 ...
2
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3answers
46 views

Why is the identity map from $S^1$ to $S^1$ not homotpic to the constant map?

Why is the identity map from $S^1$ to $S^1$ not homotpic to the constant map? I get the picture that if the identity map $id$ is homotopic to the constant map then as the circle transforms ...
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1answer
31 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
20 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
18 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
18 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 ...
2
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1answer
24 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 ...
3
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2answers
41 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 ...
0
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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
20 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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1answer
53 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.
1
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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
14 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
25 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
81 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 $C=1$, ...
3
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4answers
265 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 ...
10
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1answer
101 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 ...
4
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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
27 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
32 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 ...
2
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0answers
33 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
28 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
20 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
12 views

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
19 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
90 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 ...
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2answers
57 views

If $\overline A\cap B = A\cap \overline B = \varnothing$, $A\cup B$ is disconnected.

I'm trying to show that if $\overline A\cap B = A\cap \overline B = \varnothing$, $A\cup B$ is disconnected. First of all, I think I have to assume that $A$ and $B$ are nonempty, or else the statement ...
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1answer
45 views

A subspace of a metric space is normal

Is it true that a subset Y of a metric space X is a normal topological space? I think yes, because Y is a metric subspace of X equipped with the induced metric by the one of X. Am I wrong? Thank you ...
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0answers
27 views

Showing any metric space is a Hausdorff space

This is the question i've been given along with the solution i have written, however, could someone explain why showing $z \notin V$ shows any metric space is a Hausdorff space
2
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1answer
39 views

A counterexample for a smoth version of Tietze extension theorem

Is there any function $f:F\subset \mathbb{R}^2\rightarrow \mathbb{R}$ with $F$ closed such that $f|F$ is differentiable in every accumulation point but there is no differentiable extension to the ...