1
vote
1answer
11 views

Paradox in connection with definition of limit points and order limit theorem?

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I come across something that appears (to me) as a paradox. Let me first write down one definition and two theorems that ...
0
votes
1answer
33 views

Exercise 3.3.8 from Understanding Analysis by Stephen Abbott

Motivation: trying to prove that if $K \subseteq \mathbb{R}$ is compact (and thus, by the Heine-Borel theorem, closed and bounded), then this implies that any open cover for $K$ has a finite subcover. ...
1
vote
2answers
45 views

The boundary of an open subset of $[0,1]$ containing all rationals in $(0,1)$

If $A\subset [0,1]$ is the union of open intervals $(a_i,b_i)$ such that each rational number of $(0,1)$ is contained in some $(a_i,b_i)$, prove that the boundary (frontier) of $A$ is $[0,1]-A$. ...
3
votes
2answers
82 views

About the proof of the Heine-Borel Theorem

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I have a question about the prove of theorem 3.3.4 on page 84 (i.e. the Heine-Borel theorem). To be more specific, let us ...
5
votes
2answers
83 views

A subset of $[0,1]\times[0,1]$ containing at most one point from each horizontal and vertical section whose boundary is $[0,1]\times[0,1]$

How can one build a subset $A\subset [0,1]\times[0,1]$ containing at the most one point from each horizontal and each vertical section and whose boundary (frontier) is $[0,1]\times[0,1]$? I don't ...
4
votes
1answer
129 views

Why the whole theory about differentiable manifolds is based on open sets?

I only have studied basic topology, which means i haven't studied any about differentiable manifolds. I just skimmed pages on wikipedia. Here is a simple illustration on a basic situation. Let $E$ ...
3
votes
2answers
53 views

Deciding if sets are bounded and/or closed

How do we find out if a given set is bounded or closed? 1) $\{(x,y,z)\in \mathbb R^3 : x^2+2y^2-3z^3=1\}$ 2) $\{(x,y,z) \in \mathbb R^3 : |x|+2|y|+3|z|\le 1\}$ 3) $\{(x,y)\in \mathbb R^2 : ...
2
votes
1answer
40 views

Can every open set of $\mathbb{R}$ be written as countable union of disjoint open bounded intervals?

I know that the following two statements are correct. Every open set of $\mathbb{R}$ be written as countable union of disjoint open intervals ( including open rays and $\mathbb{R}$). Every open ...
0
votes
2answers
41 views

Continuous bijection from $[0,1]$ to $[0,0.5)\cup (0.5,1]$

Can we have a continuous bijection from $[0,1]$ to $[0,0.5)\cup (0.5,1]$?
0
votes
0answers
31 views

Find the point implied by intermediate value theorem

Consider a function $f(x)$ such that $f(0)=0$ and $$f'(x) = \frac{T-x}{T-f^{-1}(x)} + \frac{T-x}{S}$$ Then we can see that $f'(0)>1$ and $f'(T)=0$. Find $x$ such that $f'(x)=1$, in terms of the ...
5
votes
0answers
89 views

Show that it is an algebra.

This excercise is a little struggling for me. The part I need help with is showing that $D$ is closed under complements. Let $C$ denote the collection of all intervals on $\mathbb{R}$, including ...
2
votes
4answers
81 views

For an irrational number $a$ the fractional part of $na$ for $n\in\mathbb N$ is dense in $[0,1]$

How to prove that the $\{$ fractional part of $n\alpha\mid n \in \mathbb{N}$ $\}$ is dense in $[0,1]$ for an irrational number $\alpha$. NOTICE that $n$ is in $\mathbb{N}$ Also notice that this is ...
1
vote
2answers
44 views

Construction of a small but fat set? [duplicate]

Is it possible to find a subset $A$ of the real line $\mathbb R$ such that the Lebesgue measure of $A$ minus its interior is positive ?
1
vote
1answer
79 views

The map $t\mapsto (\cos t,\sin t)$ is injective from $[0,2\pi)$ onto the circle, but its inverse is not continuous

Question: given $\phi:[0,2\pi[\mapsto\mathbb{R}^2$ a map defined by $\phi(t)=(\cos t,\sin t)$ then Shown that $\phi$ is injective into unitary circle $S^1=\{(x,y)\in\mathbb{R}^2\mid x^2+y^2=1\}$ ...
1
vote
1answer
34 views

Topological space of continuous function is not compact

I'm struggling with this question: Let $C[0,1]$ be set of continuous function of $[0,1]$. Define metric $d(f,g)=\int^1_0|f(x)-g(x)|dx$. Show that $C[0,1]$, with topology $\tau$ induced by $d$, is ...
0
votes
0answers
16 views

Family of functions depending continuously on a parameter space WRT the $L^1$ norm

The material I'm reading involves a family of functions induced by a parameter space homeomorphic to an open disk. It attempts to show that the functions depend continuously on this parameter with ...
2
votes
1answer
33 views

Proving the Weierstrass M-Test with topology

I've encountered some theorems in analysis that are ultimately provable in a more elegant way with topology. So, is there a topological proof of the Weierstrass M-Test, ideally not using terribly ...
1
vote
0answers
31 views

Prove that every subset of $\mathbb{R}$ is compact in the finite complement topology.

I need help with my proof in particular. I am aware that there is a similar question elsewhere. Can someone please verify my proof or offer suggestions for improvement? Prove that every subset of ...
0
votes
2answers
46 views

Why is this a quotient map

Is there a direct way to see that $p \times id : [0,1]^2 \rightarrow S^1 \times [0,1]$ is a quotient map with $(p \times id)(x,y) = (e^{ix},y)$? By direct way, I mean is there an obvious argument why ...
0
votes
0answers
16 views

How to call a function defined on a set with gaps on arbitrarily small scales.

Let $I$ be an interval and $A\subset I$ such that for any two points $x,x'\in A$ there exists an interval $J$ between $x$ and $x'$ such that $J\cap A=\emptyset$. How does one call this proerty of ...
0
votes
1answer
47 views

Show that C is a closed convex subset and its element of minimum norm

I have a lot of problems with the following exercise that I can't solve. Let $(L^1((0,1)), \|\cdot\|_{L^1}):=(E,\|\cdot\|)$ and $$C:=\{u\in E:u\geq0 \text{ a.e } x\in (0,1),\quad T(u)\geq 1\},$$ ...
1
vote
1answer
41 views

Choice of Metric Gives Nice Topological Properties

I am looking for examples where choosing one possibility out of many for a metric gives nice topological properties compared to the other choices. Nice is defined as compact, Hausdorff, or whatever ...
2
votes
1answer
42 views

Topology of a nested sequence of subsets

Hi everyone I'd like to know if the following proof is correct, I think so. And also if there is a more direct approach without the many subcases. Thanks in advance Let $X$ be an infinite set, and ...
3
votes
1answer
84 views

Non Archimedean field

Prove that there exist an ordered field which is not complete but in which every cauchy sequence has a limit? Here Completeness means every bounded above subset has a least upper bound. And for cauchy ...
0
votes
0answers
42 views

Is Hausdorff necessary condition for Arzela-Ascoli? [duplicate]

Here is a theorem proven in Munkres-Topology Let $(X,\tau)$ be a nonempty compact space. Let $d$ be any metric induced by a given norm on $\mathbb{R}^n$. Let $\overline{\rho}$ be the ...
0
votes
2answers
24 views

Nullhomotopic map extended

I have troubles understanding this proof: Let $h:S^1 \rightarrow X$ be a continuous map, then we have that if $h$ is nullhomotopic, $h$ can be extended to a continuous map $k:B^2 \rightarrow X.$ ...
3
votes
1answer
53 views

Real analysis question about boundedness

In real analysis courses, students are often taught a theorem which states that: If $f$ is a real valued continuous function on $[0,1]$, then $f$ is bounded there and the example ...
0
votes
1answer
32 views

Fibre is open in covering space

I think I don't see the wood for the trees: In my notes I found the remark that if $p:E \rightarrow B$ is a covering map, then for each $b \in B$ we have that $p^{-1}(b)$ in $E$ has the discrete ...
-1
votes
1answer
61 views

Topological properties of $(0,1)\times \{0\}$

I am having a real hard time solving simple proofs involving open sets. I am confronted with this one: Is $(0,1)\times \{0\}$ open? Is it compact? What is its interior? I know $(0,1)$ is open. ...
1
vote
1answer
29 views

how we get a closure these set

I have some questions how we get a closure these set from any interval of real number For example (1,2),[3,6],(8,10],(2,infinity),(-infinity,4]? what is the formula closure in metric space?
1
vote
1answer
39 views

Deck transformations

We have a theorem that says that if a group $G$ acts on a path-connected space $Y$ properly discontinuously, then $\pi: Y \rightarrow Y/G$ is a covering map. Especially, $G$ is isomorphic to the group ...
1
vote
3answers
94 views

Is the function $f(x) = 1/x$ continuous?

A function f is mapped from the non-zero reals to the reals . We assume the natural topology to be induced on the domain. Then is the function f(x) = 1/x continuous ? EDIT Suppose I use this ...
0
votes
0answers
44 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: ...
1
vote
0answers
45 views

Baker's transformation: continuity, orbits of irrational and rational points

I've reading the Pugh's Analysis book and I have problems with one exercise. This says: The baker's transformation: a rectangle of dough is stretched to twice its length and folded back on itself. ...
1
vote
2answers
68 views

If a function $f:\mathbb R\to\mathbb R$ is proper, then it tends to infinity as $x\to \infty$

I came across this problem the other day, I've played around with it but still don't really have any ideas: "A function, $f$, is said to be proper if the pre-image of any compact set is compact. Show ...
2
votes
1answer
62 views

Characterization of dense open subsets of the real numbers

Does the complement of every dense open subset of the real numbers have Lebesgue measure $0$? This is certainly not a characterization of dense open subsets of reals, since the complement of the ...
1
vote
1answer
41 views

What does it mean that the set of polynomials is dense in $C^0([a,b],R)$

What does it mean that the set of polynomials is dense in $C^0([a,b],R)$ $C^0( [a,b ], R )$ is the set of continuos functions. As I understand it, for the set of polynomials (call this set $P $) to ...
1
vote
0answers
37 views

Characterising subgroup

Let $\omega $ be a path in $\hat{X}$ with $\omega(0), \omega(1) \in p^{-1}(x_0)$, where $p$ is a covering map $p:\hat{X} \rightarrow X$. Let $\alpha=[p \circ \omega] \in \pi_1(X,x_0)$. Then we have ...
-1
votes
2answers
63 views

On invertible matrix in $\mathbb R^{n^2}$ [closed]

How do i prove that the invertible matrix form an open and disconnected set in $\mathbb R^{n^2}$ or generally if $G$ its a multiplicative group of matrices in $\mathbb R^{n^2}$ with Int($G$) non ...
0
votes
1answer
52 views

Correct proof of supremum property?

Let $u$ be an upper bound of non-empty set $A$ in $\mathbb{R}$. Prove that $u$ is the supremum of $A$ if and only if for all $\epsilon > 0$ there is an $a \in A$ such that $u-\epsilon < a$. ...
3
votes
0answers
89 views

Non-trivial compatibility which makes convex functions continuous on $\Bbb R$

Here are the definitions: Let $X$ be a set. Another set $\mathcal C\subseteq \mathcal P(X)$ is called a convexity over $X$ if $\varnothing, X\in\mathcal C$ $\mathcal C$ is closed under arbitrary ...
3
votes
1answer
63 views

Lebesgue covering dimension of $[0,1]$

Say, we define the Lebesgue covering dimension (LCD) like this: A set $S\in \mathbb R^n$ has LCD $d\in \mathbb N$ if and only if $d$ is the smallest natural number such that for any open cover ...
1
vote
0answers
41 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 $t$, ...
0
votes
0answers
66 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 ...
0
votes
1answer
28 views

Fundamental group smash product

is there a way to conclude what the first fundamental group of the smash product of two path-connected spaces is? probably there should be a general way like there is for the wedge sum due to van ...
0
votes
1answer
48 views

Minkowski Distance Metric

Given compact sets $A$, $B$, define the Minkowski distance between the two sets as: $$ \delta(A,B):= \inf \{ r: B \subseteq \mathscr{N}_r (A) \, \, \text{and} \, \, A \subseteq \mathscr{N}_r (B) \}$$ ...
0
votes
1answer
42 views

Rudin Real and Complex Ch.2 question 16

This excerise 2.16 in Rudin is as follows: Let X be the plane with the following topology: a set is open iff it's intersection with every vertical line is an open subset of that line w/ respect to the ...
0
votes
0answers
23 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 ...
2
votes
1answer
50 views

The definition of Compactness for “set” and “space”

Compactness for "set" and "space" I was wondering if there is any significance between the two settings. Do we treat them as two different things? For example, let $(X,d)$ be a metric space with the ...
1
vote
1answer
25 views

Group action and set define via their quotient topology open/closed equivalence relations

If we have a topological space $X$ and a subset $A \subset X$, we can define $X \backslash A$. My question is: Is it true that this equivalence relation is closed iff $A$ is closed as a subset of $X$ ...