2
votes
1answer
29 views

Distance between a point and a closed set in metric space

Here is what I am thinking. Let (X,d) be a metric space and let C be a closed subset of X. Fix any poin p in X. Then, there exists a point q in C such that d(p,q) = distance(p,C). I think this ...
0
votes
0answers
27 views

Point-set topology - derived sets, boundaries, etc.

Consider the metric space $(\mathbb R^2,\sigma)$, where $\sigma$ is the discrete metric. Let $$E=\lbrace(x,y)\in \mathbb R^2 : x^2+y^2<1 \rbrace$$ i.e. the unit disc. Find the following sets ...
2
votes
1answer
45 views

Is every closed ball (or open ball) in the Eucledean Space $R^n$ convex?

I am solving a problem and I need to use this fact: Every closed ball (or open) in the Eucledean Space $R^n$ convex? Hoever, I am not sure if it is true or not. Can anyone help? Thanks!
5
votes
0answers
47 views

How to master general topology for analysis?

I started learning topology long ago. I first exposed myself to metric topology in Baby Rudin and Munkres Topology 2nd ed. Part I. Munkres is my most revisited book ever since. The first big ...
1
vote
1answer
29 views

What sequences could satisfy these requirements?

I need to find a sequence which converges to $0$ but is not in any space $\ell^p$, where $1 \leq p < +\infty$. And, I need to find a sequence which is in every space $\ell^p$ with $p > 1$ but ...
0
votes
1answer
32 views

Searching for analytical or topological proof(s) of the Cayley-Hamilton theorem

Is there any analytical or topological proof(s) of the Cayley-Hamilton theorem ? I want to know such proofs ( if possible ) , I would even appreciate proper references with accessible links . Thanks ...
1
vote
1answer
30 views

Where was I supposed to use compactness in this proof that a compact subspace can be separated from a point by open sets?

Can anyone check my proof below? P. Let $X$ be a metric space. Prove that if $K\subseteq X$ is compact and $x\notin K$, there exist disjoint open sets $U$ and $V$ such that $K\subseteq U$ and ...
0
votes
0answers
19 views

Properties of normal set

For any closed subset $F$ of $X\subset \mathbb{R}^{n}$, we define the normal set $\mathcal{N}(F)$of $F$ as follows: if there exists $f\in C^2(X)$, and $x_0\in F$, such that $$ df(x_0)\neq 0;\\ ...
0
votes
1answer
30 views

For $f:D\subset \Bbb R^n \rightarrow \Bbb R^m$ prove the following are equivalent:

For $f:D\subset \Bbb R^n \rightarrow \Bbb R^m$ prove the following are equivalent: a)$f$ is continuous in $D$ b)If $O\subset \Bbb R^m$$f$ is an open set, then there exists an open set $G\subset ...
1
vote
1answer
57 views

An open set in $\mathbb{R^n}$ is connected if and only if it is path connected

Here is a proof I found on the internet but cannot understand a part of it which is highlighted. I hope someone can help me understand this. Thanks in advance
1
vote
0answers
39 views

Point-set topology, characterizing sets

I think the question sort of speaks for itself. Please no solutions in the answers - I'm mostly looking to see if my logic makes sense: Let $E$ be a set in a metric space $(X,d)$. Let $E^{\circ}$ ...
4
votes
1answer
66 views

Is $d(x,y) = (x-y)^2$ a metric on $\Bbb R$?

For $x,y,z \in \Bbb R$, define $d(x,y):= (x-y)^2$ Is this a metric on $\Bbb R$? It's clear that $d(x,x)=0$ and $d(x,y)=d(y,x)$ for all $x,y \in \Bbb R$. The triangle inequality seems to have a ...
3
votes
2answers
45 views

Definition of neighborhood

I am starting to work through Rudin's Principles of Mathematical Analysis. For $(X, d)$ a metric space and $x \in X$, Rudin defines the neighborhood $N_r(x)$ of $x$ to be the set consisting of all ...
1
vote
1answer
21 views

Completion and Seprability of C[0$\infty$]

If I have C[0,$\infty$] the space of all continuous functions on [0,$\infty$] with metric $$ \phi(\omega_1, \omega_2) = \Sigma^{\infty}_{n=1} (1/2^n)*max_{0{\leq} t {\leq} ...
0
votes
1answer
38 views

Is it true that $\textrm{supp}(f)\subseteq K$ implies $f|_{\partial K}=0$?

Maybe this will be an elementary question but I need to clarify this. Let $X$ be a metric space and let $f:X\longrightarrow \mathbb R$ continuous. Suppose $\textrm{supp}(f)\subseteq K$ where $K$ is ...
2
votes
1answer
21 views

Counterexample on weaker version of result about compact sets

The following is a very well known theorem: Let X be a metric space. $K \subset X$ is compact iff every collection $ \{ F_j \}_{j\in A}$ of closed sets with the finite intersection property in K ...
0
votes
2answers
38 views

Name for “bicontinuous function” that's not bijective?

We know that an invertible continuous function whose inverse is also continuous is called a homeomorphism. But is there a name for a not-necessarily-bijective function that is "bicontinuous" in the ...
2
votes
1answer
166 views

Reference request: Analysis, Algebra and Topology - Same author(s)/publisher(s), progressive order

Is there anywhere I can acquire a collection of all Mathematical undergraduate textbooks by the same publishing author, or authors(so that they are similarly written) and can be completed in a logical ...
1
vote
0answers
43 views

Stuck on continuity proof (like 8 sheets of A4…) $p_if$ is cont. iff $f$ is cont, $p_i:X\rightarrow X_i$ given by $p_i(a)=a_i$ for $a=(a_1,…,a_n)$

Let $Y$ be a metric space, let $f:Y\rightarrow X$ where $(X,d)$ is a metric space given by $X=\prod^n_{i=1}X_i$ equipped with the stadard metric ($\max$) I wish to prove $f$ is continuous iff ...
3
votes
0answers
66 views

Condition on Equality of closure of open ball and closed ball, suppose I have a counterexample

Let $(X, d)$ be a metric space. Also for $x \in X$ and $r \ge 0$ define: $$ B(x,r) = \{ y \in X : d(x,y) < r \} \quad \mbox{ and } \quad K(x,r) = \{ y \in X : d(x,y) \le r \}. $$ Denote by ...
0
votes
1answer
71 views

Continuity, and continuity in topology.

Metric spaces: Neighborhood of a point $a$ is a Set of point $N$, such that $\exists\delta>0:B_\delta(a)\subset N$ ($B_r(x)$ = open ball at x of radius r) Definition of open set: "A subset $O$ 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
3answers
51 views

Closure of a subset of a metric space is closed

From definition, if $X$ is a metric space, if $E \subset X$, and if $E'$ denotes the set of all limit points of $E$ in $X$, then the closure of $E$ is the set $\overline{E}=E \cup E'$. I need to ...
3
votes
1answer
56 views

Disconnected Topoological Space with Intermediate Value Property

Does There exist a disconnected topological space with intermediate value property? Intermediate Value Property states that 'a topological space X is said to have intermediate value property if for ...
0
votes
2answers
27 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 $k:B^2 \rightarrow X.$ ...
8
votes
4answers
1k views

Can a set be infinite and bounded?

I don't understand a statement in my math book course, I was restudying the compact sets part of the chapter when at a certain moment there is a corollary saying : 'every infinite and bounded part of ...
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
62 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. ...
2
votes
0answers
31 views

Isomorphism isometries between finite subsets , implies isomorphism isometry between compact metric spaces

Let's $(X_1,d_1), (X_2,d_2)$ be compact metric spaces such that for every finite subset of $X_1$ like $A$ (respectively any finite subset of $X_2$ like $B$ ) there exists a finite subset of $X_2$ ...
1
vote
1answer
46 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
98 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 ...
6
votes
1answer
75 views

Show that there is sequence of homeomorphism polynomials on [0,1] that converge uniformly to homeomorphism

Let $f:[0,1]\rightarrow [0,1]$ be a homeomorphism. Show that , there exists a sequence of polynomials $$(P_n(x))_n$$ such that $P_n(x)$ converge uniformly to $f$ on $[0,1]$ and every $P_n(x)$ is a ...
-1
votes
1answer
68 views

Topology question about open spaces of a topological space homeomorphic to the full set. [closed]

Let $\mathcal{U}$ be an open subset of $\mathbb{R}^m$ such that there is homeomorphic $f$ from $\mathcal{U}$ to $\mathbb{R}^m$ and also $f$ is an uniformly continous function. Show that ...
0
votes
1answer
29 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
0answers
24 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 ...
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 $X$ ...
0
votes
1answer
32 views

Fundamental polygon

So, I have seen fundamental polygons quite a few times now and I was always wondering what they are actually good for. Let's take the sphere. It's fundamental polygon can be seen here image. Does ...
5
votes
1answer
70 views

Visualisation of the smash product

wedge product, join etc. all of them are no problem for my head, but I am really failing to get a visual idea of what the smash product wants to tell me. For example if I take two spheres, I have no ...
2
votes
1answer
43 views

Proof of a distance

I have one distance shown as an example in a book but I'm striving to demonstrate that it is effectively a distance. here it is said : let $U=\{z\in\mathbb{C, |z|=1}\}$ we can get a distance on $U$ ...
1
vote
1answer
43 views

The existence of $f \in C^\infty(R^n)$ with $ f=0$ on closed $E$, otherwise $f>0$

This is problem 6.3 in 'Rudin's Functional analysis If $E$ is an arbitrary closed subeset of $R^n$, show that there is an $f \in C^\infty(R^n)$ such that $f(x)=0$ for every $x \in E$ and ...
1
vote
0answers
29 views

Conjunctive Normal Form representation/ First Order Logic.

in my research problem, I need to represent three types of three types of relationships between the variables x,y as the following:: " y Cooperates with x" relationship: means if there is two ...
0
votes
1answer
57 views

Density and convergence

I have a small question: Is it true that if the basis of a space $A$ is dense in a space $B$ ($B\subset A$) then if $u_n\rightarrow u$ in $A$ we have that $u_n\rightarrow u$ in $B$ ?
1
vote
3answers
114 views

Suggestions for a real analysis reference.

Can anyone suggest some real analysis book which has a geometric presentation of the concepts with pictorial representation.
0
votes
1answer
38 views

Proof of an open set or closed set

I'm struggling on a proof that I can't proof correctly. Let $A=\mathbb{Z}$, $B=\{n-\frac{1}{2n} | n \in \mathbb{N}*\}$ I could prove easily that A is a closed set and B as well : $\overline A =$ ...
1
vote
3answers
111 views

Why are open sets used in definitions in differential geometry?

I find that in most definitions in differential geometry, such as those of defining a manifold, a smooth manifold,differentiable functions, diffeomorphisms on manifolds, an atlas,etc , open sets are ...
1
vote
1answer
35 views

Prove that the a modified Cantor Set is not Jordan-Measurable

Let $C_0 = [0,1]$ and if $C_n$ is given as a disjoint union of intervals, construct $C_{n+1}$ by removing from each interval $I$ an open interval of length $(n+2)^{-2}|I|$ in the middle of each ...
0
votes
2answers
51 views

If $a$ is a limit point of $f^{-1}(b)$, then the linear mapping $f'(a)$ is not injective.

Let $U\subset\mathbb{R}^m$ be an open set and $f:U\to\mathbb{R}^n$ a differentiable function. Suppose that there exists $b\in\mathbb{R}^n$ and $a\in U$ such that $a$ is an accumulation point of ...
0
votes
1answer
30 views

Neighbourhood of a disc

I'm a bit confused on how to write down precisely a neighborhood on an example. My question is the following: Suppose I have a disc $\Omega=\lbrace x\in\mathbb{C}, |x-1|<2.5\rbrace$ and its ...
3
votes
1answer
86 views

What are the $n$th roots of the identity function?

What are all the functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f^n=I$ where $f^n$ denotes the composition $f\circ f\circ f\dots \circ f$ of $f$ with itself $n$ times, and ...
3
votes
1answer
92 views

Real analysis with a non-standard topology

I have recently undertaken a self study of topology and am using Munkres Topology 2nd edition as the primary text. My background(theoretical chemistry & physics) is almost entirely void of any ...