Tagged Questions
1
vote
2answers
56 views
Is this function from $[-1,1] \rightarrow \mathbb{R} \cup \infty$ continuous?
just a short question. So I was wondering about functions from compact sets into
$\mathbb{R} \cup \{+\infty\}$.
Let's say we have a function $f : [-1,1] \rightarrow \mathbb{R} \cup \{+\infty\}$,
...
9
votes
4answers
218 views
What will be a circle look like considering this distance function?
I am working on some exercises in the book Geometry: A Metric Approach with Models by R.S. Millman. He defines the following map: $$d_S(P,Q):\mathbb R^2\times\mathbb R^2\to\mathbb R\\\ ...
2
votes
1answer
58 views
About measure theoretic interior and boundary
Reference:- Evans-Gariepy, Federer, other books and notes of geometric measure thoery.
I just want to clarify whether these definitions of measure theoretic interior
and boundary are correct. Given ...
6
votes
1answer
81 views
Regular open set whose boundary has nonzero volume.
I found this question quite interesting, but its answers were disappointingly non-geometric. I'd be interested to know whether there exists a geometric example.
To be precise about what I mean by a ...
4
votes
3answers
84 views
Domain whose boundry has non zero volume.
Can There be a domain in $\mathbb{R^n}$, for any $n$ such that some domain has non zero boundry volume? I.E. volume of boundry is non zero?
Motivation:
In some theorems, it is specified that volume ...
3
votes
3answers
132 views
Topological boundary vs geometric boundary
Let $M_1=B((0,0),1)=\{(x,y) \mid x^2+y^2<1\}$
$M_2=\{(x,y) \mid x^2+y^2\le1\}$
What are the interior of $M_1$ and $M_2$ ?
And What are the boundary of $M_1$ and $M_2$ ?
How to find them? ...
1
vote
2answers
45 views
Parametrization of $n$-spheres
This comes from Guillemin and Pollack's book Differential Topology. The book claims that one cannot parametrize a unit circle by a single map. I thought we could (by a single angle $\theta$).
I ...
4
votes
1answer
40 views
Sum of Cauchy sequences in an abelian topological group (first countability hypothesis)
We know that, given a first countable abelian topological group $G$, the sum of two Cauchy sequences gives yet another Cauchy sequence (see, e.g., this answer).
For those wondering, we say that a ...
3
votes
1answer
34 views
Is a continuous function in two variables necessarily equicontinuous in one variable?
Suppose $K \in \mathcal{C}\left(\left[0, 1\right]\times\left[0, 1\right]\right)$. Then, is it necessarily the case that the set of functions $\left\{g_y(x):g_y(x) = K(x,y), \forall y \in ...
2
votes
3answers
58 views
Basic topology question regarding the complex plane.
Prove that the Complex plane is closed, open and perfect.
My intuition is destroyed by the fact that a set can be open and closed at the same time.
The following is my understanding.
open: If all ...
3
votes
2answers
54 views
A minor question about the Cantor Set
I'm self teaching analysis and the second chapter is about some basic topology.
According to the book "Principles of Mathematical Analysis (3rd)" from Walter Rudin,
the Cantor Set is constructed as ...
1
vote
1answer
26 views
On the limit of a Minkowski sum
Consider an open set $\mathcal{O} \subseteq \mathbb{R}^n$.
I am wondering if the set $$ \mathcal{S} := \lim_{k \rightarrow \infty} \ \mathcal{O} + \frac{1}{k} \mathbb{B} $$
is open or closed.
With ...
2
votes
3answers
47 views
Klein Bottle discrete harmonics?
Studying discrete representations of a function, on a $2$ Dimensional compact surface, brings to the use of spherical harmonics for the 2-sphere, and discrete Fourier transformations for the 2-torus.
...
5
votes
0answers
33 views
$M$ is compact, non-empty, perfect, and $M \cong M \times M$. Must $M$ be homeomorphic to the Cantor set, the Hilbert cube, or some combination?
Assume that $M$ is compact, non-empty, perfect, and homeomorphic to its Cartesian square, $M \cong M \times M$. Must $M$ be homeomorphic to the Cantor set, the Hilbert cube, or some combination of ...
1
vote
0answers
27 views
Show that an induced map on a quotient space is a homeomorphism.
Consider the unit square $Q := [0,1] \times [0,1]$ with the coordinates $(x,y) \in Q$. Let $Z := Q/\thicksim$ be the quotient space, which results by identifying:
\begin{align*}
(x, 0) & ...
2
votes
1answer
33 views
Boundedness In Fractal Space
Let $\mathbb H$ be family of all non empty compact subsets of $\mathbb R^n$ and $d_H$ be Hausdorff distance on $\mathbb H$. Some called this $(\mathbb H,d_H)$ as Fractal Space. Is boundedness equal to ...
2
votes
1answer
84 views
spectrum of two bounded linear operators
Suppose that L and B are bounded linear operator on H, assume $0\in \rho(L) \cap \rho(L+B)$ and that $L^{-1}$ is compact. Prove that L+B also has a compact inverse.
0
votes
1answer
42 views
Minkowski functional satisfies triangle inequality on convex sets
New here so apologies if I screw up any decorum.
Taking analysis, have a problem set that's tilting toward topology. The problem asks for proof that the Minkowski functional associated with a set K ...
1
vote
0answers
33 views
The Lebesgue number property and uniform continuity (proof check)
Theorem If $f$ is continuous on a compact metric space $X$, then $f$ is uniformly continuous on $X$.
Proof Let $\epsilon>0$. For any $y\in X$ there is a $\delta_y$ such that $d(x,y)<\delta_y$ ...
2
votes
1answer
79 views
Why is the graph of a continuous function to a Hausdorff space closed?
Say I have two top. spaces given by $(X,\mathscr{T}_x)$ and $(Y,\mathscr{T}_Y)$ where $Y$ is Hausdorff. In addition say I have a functon $f:X\rightarrow Y$, and let it be continuous. I want to show ...
1
vote
2answers
45 views
Necessary and Sufficient Condition for two metrics to have same open sets.
There are couple of independent conditions like one being scalar multiple of another, or if $$d_p(x,y)=(x^p+y^p)^{1/p}$$ then all $d_ps$ and $d_qs.$ which guarantee that open sets are same under these ...
1
vote
1answer
48 views
topology in R infinity
What does the following sentence mean and why is that true:
"The nonnegative orthant in $R^{\infty}$ has empty interior in product topology"
Thank you!
2
votes
1answer
61 views
A closed form for a particular topology.
I am trying to find some sort of 'closed form' (if possible) of a particular topology generated by the sets: $({x\in\mathbb{R}\ \vert x\geq a}), a\in \mathbb {R}$.
Thanks !
0
votes
2answers
38 views
Closed and open sets
By regarding the real numbers with their natural topology, my textbook says, that:
$$ \left\{2 \pi n+\frac{1}{n}\;\bigg|\;n \in \mathbb{N} \right\}$$
is closed, which i understand, as every sequence ...
1
vote
2answers
50 views
Hausdorff space and Cantor's intersection theorem
$X$ is a Hausdorff space, $C_i$ is a non-empty closed subset of $X$ and $C_{k+1}\subseteq C_k$ , show that $\displaystyle \bigcap_{i\in \mathbb{N}} C_i$ is compact.
I tried to prove by ...
0
votes
3answers
63 views
How to understand both closed and open set in topology?
We've defined the connectedness in topology in class in this way that a topological space is connected if the only both open and closed set is empty set or the whole set.
Now I got the explanation ...
1
vote
1answer
35 views
Product of Sets
I have a quick question regarding the interpretation of notation in topology. My notes state:
Let $X:=\Pi_{\alpha \in A} X_\alpha$ where A is an indexed set.
My interpretation is that $X=X_{\alpha ...
2
votes
0answers
74 views
The Cantor Space and open, but not closed sets.
consider the space $\{0,1\}^{\mathbb{N}}$ of all infinite binary sequences, called the Cantor-Space. This space is metrizable with metric
$$
d(u,v) = 2^{-(r-1)} \qquad \textrm{ where } r = ...
1
vote
1answer
56 views
Looking for a “prime-ish” family of subsets
Is there a nontrivial (what I mean is below) example of a compact Hausdorff space $X$ and a family $\mathscr{F}$ of subsets of $X$ with the following pair of properties?
$\mathscr{F}$ is ...
2
votes
1answer
91 views
How to prove a topologic space $X$ induced by a metric is compact if and only if it's sequentially compact?
A topological space $X$ is called sequentially compact if every sequence of points in $X$ has a subsequence that converges to a point in $X$. I know it's very similar to Bolzano–Weierstrass theorem ...
0
votes
4answers
87 views
Counter-examples of homeomorphism
Briefly speaking, we know that a map $f$ between 2 topological spaces is homeomorphic if $f$ is a bijection and the inverse of $f$ and itself are both continuous.
So, can anyone give me 2 counter ...
2
votes
2answers
71 views
Homeomorphic subset of $\mathbb R^n$ such that… is actually $\mathbb R^n$
Let $ f:A\subset \mathbb R^n \to \mathbb R^n$ be a homeomorphism, such that $f$ is also uniformly continuous, and where $A$ is an open subset of $\mathbb R^n$
Prove that $A=\mathbb R^n$.
I have the ...
5
votes
2answers
155 views
Why is $\partial\partial M=\varnothing$?
Why is the border of the border of an oriented differentiable $n$-dimensional Manifold $M$ empty, that is $$\partial\partial M = \emptyset?$$
7
votes
1answer
59 views
Is $\mathcal{C}([0,1])$ homeomorphic to a Hilbert space?
Let $\mathcal{C}([0,1])$ the Banach space of continuous functions from $[0,1]$ to $\mathbb{C}$. The norm on $\mathcal{C}([0,1])$ is $f \mapsto \| f\|_{\infty}= \sup_{x \in [0,1]} |f(x)|$.
Is it ...
3
votes
2answers
48 views
Proof on showing whether the complement of the set $\mathbb Q\times\mathbb Q$ is open or close set
The question is as follows:
Given set A = $\mathbb Q\times\mathbb Q$
Determine whether the complement set $ A^c$ is open or close.
Here are my thoughts :
1/ $\ A^c = \{\mbox{all points ...
1
vote
2answers
39 views
Showing a set is closed with sequences
$G \subset R^n$
Let $\{x_k\}_{k\in N}$ a convergent sequence in $G$, ($x_k \in G$ for every $k$). $lim_{k \to \infty} x_k = a$ lays in $G$. Show that G is a closed set.
Help please, I have already ...
1
vote
1answer
49 views
How to show a topological space is compact and if it's from a metric or not?
We have $X$ is an infinite set and $\tau=\{U\subset X:X \backslash U$ is finite or is all of $X \}$. Till now, I have proved that $\tau$ is a topology and is not Hausdorf, but how can I show if it's ...
0
votes
0answers
29 views
Example of Homeomorphism [duplicate]
Can anyone give me an example of a continuous bijective map between 2 path-connected topological spaces, which is not a homoemorphism? Thanks!
4
votes
0answers
98 views
The Cantor Space as $\{0,1\}^{\mathbb{N}}$ and as $[0,1]$.
The Cantor-Space is defined as the space of all infinite binary sequences, i.e. the space $\{0,1\}^{\mathbb{N}}$. It has a natural metric,
$$
d(x,y) = \inf\{ 2^{-|w|} : w \in pref(x) \cap pref(y) \}
...
1
vote
4answers
36 views
Measure nonzero implies dense on a rectangle
This would be a very handy lemma for me but I have been unable to prove it thus far.
If $S \in \mathbb{R}^n$ is bounded and is not of measure zero, then there exists a rectangle $R$ such that $S$ ...
1
vote
1answer
65 views
Intermediate value property and closure of rational level sets implies continuity
Suppose $f$ satisfies the intermediate value property, i.e. if $f(a)<c<f(b)$, then there exists $a<x<b$ such that $f(x)=c$ and for every rational $r$, $S_r$ such that $f(x)=r$ is a closed ...
3
votes
2answers
72 views
Defining a metric space
I'm studying for actuarial exams, but I always pick up mathematics books because I like to challenge myself and try to learn new branches. Recently I've bought Topology by D. Kahn and am finding it ...
1
vote
1answer
62 views
Correctness of topological reasoning (interior, closure and boundary of sets)
I have a set $S= \{(x, y) | x\in\mathbb{Q} \land y\in\mathbb{R}\}$. I am pretty sure I am correct, however please comment on my reasoning. I do not want to be terribly formal (proving each statement) ...
2
votes
1answer
18 views
Bounded-ness of range
We are give a sequence of the form $x_n = 1/n$ in the complex metric space.
This sequence of course has a limit at 0, the range is clearly infinite, however, it said that the sequence is bounded.
...
3
votes
1answer
70 views
Intuition behind compact subspaces of a metric space
I've read up on compactness in a metric space and have found a few definitions (let $X$ be a metric space and $E \subset X$ in all the following):
$E$ is compact in $X$ if for every open covering of ...
2
votes
2answers
103 views
If $x\mapsto \| x\|^2$ is uniformly continuous on $E$, the union of all open balls of radius $r$ contained in $E$ is bounded $\forall r > 0$
A subset $E$ contained in $\mathbb{R}^n$ is such that the function $x \mapsto \left\Vert x\right\Vert^2$ is uniformly continuous on $E$. For $r > 0$, let $E_r$ denote the union of all open balls ...
12
votes
2answers
131 views
A non-compact topological space where every continuous real map attains max and min
Today I learnt in class that if $X$ is compact then any continuous map $f:X\to\mathbb{R}$ attains max and min. I was thinking if the converse is true:
If every continuous map $f:X\to\mathbb{R}$ ...
11
votes
3answers
142 views
Fundamental Theorem of Algebra for fields other than $\Bbb{C}$, or how much does the Fundamental Theorem of Algebra depend on topology and analysis?
When proving the Fundamental Theorem of Algebra, we need to appeal to analytic and/or topological properties of $\Bbb{C}$ and $\Bbb{C}[z]$. Is this going to be necessary in general, and if so, to what ...
3
votes
1answer
59 views
Smooth approximation of characteristic function of a bounded open set
Let $U$ be an open bounded set of $\mathbb{R}^n$. Is it possible to approximate $\chi_U$ as almost everywhere limit of increasing sequence of smooth functions?
0
votes
2answers
40 views
Definition of a covering and how it applies to the following example
Let $S= \{(x,y): x,y > 0\}$. The collection $F$ of all circular disks with centers at $(x,x)$ and radius $x$, for $x>0$, is a covering of $S$. Then all disks such that $x$ is rational is ...
