0
votes
2answers
20 views

Example of $x$ being adherent point but not accumulation point?

So I was just reading Apostol and I see that if $x$ is an accumulation point of set $S$, it has to be an adherent point as well. I guess it's possible for $x$ to be an adherent point only, not an ...
0
votes
1answer
17 views

Let $(X,d)$ be a metric space, $C$ a compact subset, and $K$ a closed subset. Prove that $K \cap C = \emptyset$ iff $d(K,C) > 0$.

Let $(X,d)$ be a metric space, $C$ a compact subset, and $K$ a closed subset. Prove that $K \cap C = \emptyset$ iff $d(K,C) > 0$. For this problem I was going to consider $d(x,F) = \inf d(x,y)$ ...
4
votes
0answers
16 views

Is every connected subset of the Sierpiński triangle arcwise connected?

I think this should be true. If it's indeed the case, it seems like this should be a known result, so references are welcome. I managed to prove that (assuming $S$ is the connected subset) $S$ ...
3
votes
2answers
85 views

Proving a set is compact - Homework

Let $(X,d)$ be a metric space and let {$p_n$} be a sequence of points in $X$ with $\lim_{n\to ∞}p_n = p_0$. Prove that the set $K =$ {$p_0, p_1, p_2,...$} is a compact subset of $X$. I have ...
0
votes
1answer
27 views

Any open set shares boundary with a discrete set

Claim: Let $X$ be a metric space and let $U\subset X$ be open. Then there exists a discrete set $A\subset X$ such that $\partial A = \partial U$. Approach thus far: Since this statement is about ...
1
vote
2answers
50 views

Find sets of points, where function from one topological space to another is continuous.

We have got two functions : $f(x,y) = (2x,y)$ $g(x,y) = (x+1,y) $ They are transormations from one topological space to another ( from $ (\mathbb{R^2}, \tau')$ to $ (\mathbb{R^2}, \tau'')$ ), ...
1
vote
1answer
44 views
+350

Having difficulties showing the triangle inequality of metric in the plane

Let $P \in \mathbb{R}^2$ and define $$ d(x,y) = \left\{ \begin{array}{lr} ||x-y|| & if \; \; x,y,P \; \; \text{Are Collinear}\\ || x - P|| + ||y-P||& \;\;\;\; otherwise ...
2
votes
1answer
47 views

Closed sets in a subspace are formed by intersecting the subspace with closed sets

Let $X$ be a metric space and let $Y$ be a subset of $X$ be a subspace with the induced metric. (induced means the metric restricted to elements of $Y$) Let $A$ be a subset of $Y$. Prove the following ...
2
votes
1answer
17 views

Existence of maximizer implies compact? [duplicate]

I know that compact sets imply the existence of a maximizer, but is the converse true: Let $(X,d)$ be a metric space. Suppose that whenever $f$ is a continuous (and real) function on $X$, there ...
2
votes
1answer
42 views

MAth proof questions Open closed sets

Let $X$ be a metric space and let $Y$ be a subset of $X$ be a subspace with the induced metric. (induced means the metric restricted to elements of $Y$) Let $A$ be a subset of $Y$. Prove the following ...
1
vote
3answers
31 views

Topology-Open Sets of a Metric Space

Let $(X_i,d_i), i=1,2,\dots,n$ be metric spaces. Let $X=\prod_{i=1}^{n}X_i$ and let $(X,d)$ be the metric space defined in the standard manner. For $i=1,2,\dots,n$, let $O_i$ be an open subset of ...
4
votes
1answer
29 views

What is the maximal size of an equal-distance set in $\mathbb{R}^n$?

Let $A\subseteq \mathbb{R}^n$ with the casual metric and $c\in\mathbb{R}^+$ be a real positive number, such that for every $a_1, a_2\in A$ if $a_1\neq a_2$ then $d(a_1,a_2)=c$. What is the maximal ...
2
votes
1answer
22 views

Why open unit ball in any infinite dimensional Banach space is finitely chainable?

In paper "Pointwise products of uniformly continuous functions" by Sam B. Nadler, Jr., He defined the finitely chainable as followings : Let $(X,d)$ be a metric space. An $\varepsilon$-chain in ...
1
vote
1answer
18 views

Approximation of acontinuous function

How to approximate a continuous function on $[-\pi,+\pi]$ which is $2\pi$ periodic by a set of trigonometric polynomials in the sup-norm topology?
1
vote
1answer
45 views

Metric Space and Open Sets

I'm having trouble figuring out where to go with this problem. Any hints or strategies would be appreciated. I have just the basic definitions for open sets, distance metrics, etc. Consider $\Bbb ...
1
vote
1answer
14 views

Continuity of function and its value.

Here's a problem I'm struggling with. Not really sure how to do this. My tools are epsilon delta proofs for continuity and that's about it. Let $f:[0,\infty)\to\Bbb R$ be a function which is ...
3
votes
2answers
163 views

Homeomorphic metric spaces

I want to examine if $(0,1] $ and $\mathbb R $ are homeomorphic. We work on metric space $(\mathbb R, e)$, where $e$ stands for the euclidean metric. My answer: Let's assume there is a ...
1
vote
1answer
55 views

Continuity of a function?

Let $f: (\mathbb N, e_\mathbb N) \to(\mathbb R,e)$, where $e$ stands for the euclidean metric and $$f(n)=\begin{cases} n,\, n\ge 2\\\\0,\, n=1\end{cases}$$ Is $f$ continuous? Firstly, I can ...
2
votes
1answer
31 views

Topological Vector Space not induced by Metric

Can anyone give me an example of a Topological Vector Space that is not metrizable? I know that the neighborhood base of $0$ needs to be incountable, and all I can construct then is no topological ...
0
votes
1answer
43 views

continuous function from one metric space to another metric space

Is differentiation $f(x) \rightarrow f'(x)$ a continuous function from $C^1[a,b] \rightarrow C[a,b]$ ? Is integration $f(x) \rightarrow \int_a^x \! f(t) \, \mathrm{d}t $ a continuous function from ...
2
votes
2answers
54 views

Topologies generated by a metric

Hi I am new to mathematical proofs/notation and am working through John Lee's Introduction to Topilogical Manifolds. This is the question and my attempt. This is not homework. 2.4 Suppose $M$ is a ...
2
votes
2answers
42 views

topological properties of a given set

Let us consider the set $X=C[0,1]$ with its sup-norm topology. Let $S $ be the set of all elements $f$ of $X$ such that $\int_0^1 f(t) dt=0$. Is $S $ compact and connected? To show $S$ compact I have ...
2
votes
3answers
46 views

Density of sets

I have got a problem on whether a set is dense or not but not quite sure on how to approach it. Consider the space $M_2(R)$ with its usual topology.Consider the set $ S$ of all matrices with both ...
1
vote
1answer
34 views

Show that there is a series in R^infinity has some term greater than or equal to 1/n but that also is arbitrary close to the zero sequence.

Consider $\mathbb{R}^\infty = \{(a_n): \sum_{n = 1}^{\infty} a_n^2 < \infty\}$ with the metric $d((a_n$), ($b_n$)) = $[\sum_{n=1}^{\infty} (a_n - b_n)^2]^{1/2}$. Let $A = \{(a_n) : |a_n| < 1/n ...
4
votes
0answers
43 views

Closed or open subsets of $C[a,b]$?

$C[a,b]$ denotes the space of continuous real-valued functions on $[a,b]$. The metric associated with $C[a,b]$ here is $d(f,g)=sup[|f(x)-g(x)|]$ where the supremum is taken over $[a,b]$. $C^1[a,b]$ ...
1
vote
2answers
79 views

Abstracted Metric and Measure Spaces

As I am just beginning to study general topology and metric spaces in more and more detail, it seems to me that the metric space topology is entirely determined by the properties of $\Bbb R$, since ...
0
votes
0answers
17 views

Trigonometric polynomials are dense

Is the set of all trigonometric polynomials in the space of continuous functions on [$-\pi,\pi]$ which are $2\pi$-periodic dense?(with sup-norm topology)Please give hints on how to find a sequence of ...
1
vote
1answer
54 views

Metric topology induced by the sum of two metrics

I have to show the following: Let $X$ be a set with metrics $d_1$ and $d_2$ inducing metric topologies $\tau_1$ and $\tau_2$. Define a new metric on $X$ where $d(x,y) = d_1(x,y) + d_2(x,y)$ for ...
0
votes
2answers
32 views

In a metric space with a countable base, how does every open cover has a countable subcover?

Let $X$ be a mertic space, and let $\{V_{\alpha}\}$ be a collection open subsets of $X$ such that, for every $x \in X$ and for every open set $G \subset X$ with $x\in G$, there is some $V_\alpha$ such ...
0
votes
1answer
35 views

Every infinite subset of E in R having a limit point in E implies E is closed and bounded

Every infinite subset of E in R having a limit point in E implies E is closed and bounded. Could you please help with a formal proof of this result ?
0
votes
1answer
44 views

Why are $\{0\}$ and $\{1\}$ open subsets of the discrete metric space $\{0,1\}$?

$\{0\}$ and $\{1\}$ open subsets of the discrete metric space $\{0,1\}$. Let $S$ be a metric space. Then a subset $A \subseteq S$ is considered open if $\forall x \in A, \exists r>0$ such that ...
2
votes
2answers
231 views

A question on Rudin's Book “ Principles of Mathematical Analysis”

On Theorem 2.27 (a), page 35, Rudin's proof is incomplete. If it is not the case $E' \subset E$ then the proof is false. Are you agree with this observation? (The theorem is: “If $X$ is a metric ...
0
votes
1answer
37 views

Let D be the usual Euclidean metric on M. Are the two identity functions $I_1$ and $I_2$ continuous functions where d is the Manhattan metric on M?

Let D be the usual Euclidean metric on M. Are the two identity functions $I_1:(M,D) \to (M,d)$ and $I_2:(M,d) \to (M,D)$ continuous functions where d is the Manhattan metric on M? The definition of ...
2
votes
2answers
47 views

Example of closed, non bounded set in R^2

I am supposed to give an example of a closed set that is not bounded in $\mathbb{R}^2$. My idea was the graph of $y=1/x, \forall x$. If I take the complement of it, I get an open set. So the graph of ...
1
vote
1answer
50 views

Let $X$ be a metric space. If $A ⊂ X$ is a compact set, prove that for any open covering, there exists a countable subcovering.

Let $X$ be a metric space. If $A ⊂ X$ has the property that every infinite subset of A has an accumulation point in $A$, show that for any open covering of $A$, there exists a countable subcovering. ...
0
votes
1answer
24 views

set of continuous functions to continuous functions: is $R$ complete?

Hi, I can do part (i) and (ii) but have trouble understanding part (iii). I can't intutively feel what the map R does. It takes continuous function to continuous function? How would I start the ...
0
votes
1answer
41 views

Topology, metric spaces, equivalence of metric spaces

The open $n$-cube is the set of all points $x=(x_1,x_2,\dots,x_n)\in\mathbb R^n$ such that $0<x_i<1$ for $i=1,2,\dots,n$. Prove that the open $n$-cube, considered as a subspace of $(\mathbb ...
1
vote
1answer
32 views

Metric equivalence

Let $ (X,d_1)$ and $ (Y,d_2) $ be two metric spaces. Define a one to one function $ f : X\to Y $. Define a new metric on $ X$ as $ d'(x_1,x_2) = d_2(f(x_1),f(x_2)) $. Question 1) Are $ d_1 $ and $ ...
3
votes
1answer
49 views

Is the space $B([a,b])$ separable?

Let $a$, $b$ be two real numbers such that $a < b$, and let $B([a,b])$ denote the metric space consisting of all (real or complex-valued) functions $x=x(t)$, $y=y(t)$ that are bounded on the closed ...
-1
votes
2answers
38 views

Compact set example

Can you please give me an example of a set that is closed but not compact in R^2\Bbb? I know that a compact set is the one that is closed and bounded, and the set [a,b] is compact. But this question ...
0
votes
1answer
22 views

Characterization of Discrete Sets in R

Let A be a subset of $\Re$ . Does anyone have a characterization of discrete sets A ( which only have isolated points ) ? I'm coming up with A is discrete iff ( A is finite) or (A is infinite and ...
10
votes
0answers
117 views

Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
0
votes
3answers
39 views

existence of unique fixed point

Let $(X,d)$ be a compact metric space and $f:X \to X$ satisfies $d(f(x), f(y))< d(x,y)$ for distinct $x$ and $y$. Then, show that $f$ has a unique fixed point. I tried this question by formulating ...
2
votes
1answer
48 views

continuity and closure questions - topology

Let $(X,d)$ be a metric space. Let $U \subseteq (X,d)$. let $k \in (X,d)$. Prove that if $U$ is fixed, $d(U,k)$ is a continuous function of $k$. Prove that $\overline{U} = U \cup V$ where $V$ is the ...
1
vote
0answers
36 views

Is there any standard procedure to properly define a composite metric?

For example, space $A$ has a metric $\rho$, and its subspace $B\subset A$ has a metric $d$, which happens to have much better properties than $\rho$. So if $x_{1},x_{2}\in A\setminus B$, but they are ...
1
vote
4answers
58 views

How to give a rigorous proof of this fact about closures of open balls in the euclidean spaces?

Let $n$ be a positive integer, $\vec{a} \in \mathbb{R}^n$, and $r > 0$. Then it is intuitively clear that the closuer of the open ball $$B(\vec{a} ; r) \colon= \{ \vec{x} \in \mathbb{R}^n \colon ...
3
votes
3answers
44 views

Showing $f$ is continuous on $M$ if $M=\bigcup_{n=1}^{\infty} U_n$

Let $f:(M,d)\to (N,\rho )$. If $M=\bigcup_{n=1}^{\infty} U_n$, where each $U_n$ is open, and if $f$ is continuous on each $U_n$, show that $f$ is continuous on $M$. Attempt: I note that ...
-1
votes
1answer
30 views

On Pseudometric

How a pseudometrics induces topology? Can anyone discuss on this topic or give any good reference?
0
votes
1answer
37 views

Constructing a countable dense subset of a totally bounded set

Given a metric space $(X,d)$, and (non-empty) totally bounded set $E$ in $X$, is it possible to construct $D \subseteq E$ which is countable and dense? I feel that this should definitely be possible. ...
1
vote
1answer
36 views

proof that a set of all bounded real valued functions is complete.

I am trying to understand the proof below. I know that a set A is complete if all Cauchy sequences converges in A. I don't understand 7th line of the proof. Why do we consider particular $x_0 \in X$ ...