0
votes
4answers
43 views

Topological spaces X and Y and a continuous bijection $f : X → Y$ while $f^{-1} : Y → X $ is not continuous

Give an example of topological spaces X and Y and a continuous bijection $f : X → Y$ such that $f^{-1} : Y → X $ is not continuous.
1
vote
1answer
24 views

The interval $(0,\infty)$ is an open set.

I want to prove this using interior points, $\epsilon$-neighborhoods and interior sets. The interior of a set A is denoted $A^o$. To show that $(0,\infty)$ is an open set, we must show that ...
1
vote
1answer
19 views

Ambiguity definitions - accumulationpoint

The literature is a bit ambiguous in my point of view. Limit points and accumulation points seems to be the same. I can accept that; that's just two names for the same. But I've seen different ...
0
votes
2answers
15 views

Cantor Intersection Theorem in $R^n$

I am looking at the Cantor Intersection Theorem from Apostol's Mathematical analysis. Let {$Q_1, Q_2, $...} be a countable collection of nonempty sets in $R^n$ such that: 1) $Q_{k+1} \subset Q_k$ ...
3
votes
1answer
22 views

Is there a subset of R such that their Cantor-Bendixson rank is the first limit ordinal?

I'm looking for a set $A \subset \mathbb{R}$ such that $\bigcap^\infty_{n=0} A^{(n)} $ is a perfect set (i.e $X'=X$) but $\forall n \in \mathbb{N}$ the set $A^{(n)}$ isn't perfect (where ...
1
vote
2answers
48 views

Limit of a sequence and a closed set

It's a dumb question, but I need to assure myself: If $V$ is a subset of a metric space $W$, then if we take a sequence in $V$ and it has a limit in $W\setminus V$, does it mean that $V$ is not ...
0
votes
1answer
28 views

If a subset $E$ of $R^n$ is bounded then E is totally bounded

I am trying to prove the above proposition. The book that I am looking at contains E in a cube of the form T=[−b,b]×⋯×[−b,b] for some large b>0. Then, since any subspace of a totally bounded metric ...
2
votes
0answers
36 views

A subset E of $R^n$ is totally bounded if and only if E is bounded

I am studying Compactness in metric space with Gamelin and Greene's Introduction to Topology and am confused about lemma 5.4 in the book. A metric space $X$ is totally bounded if for each $e > 0$, ...
1
vote
0answers
30 views

Sequence of increasing compact sets

Suppose $X$ is a locally compact metric space which is $\sigma$-compact. Let $K$ be a compact subset of $X$. We can find a sequence of compact sets $K_{n}$ such that $K_{n} \subset \textrm{int}(K_{n + ...
1
vote
1answer
29 views

Show that any infinite set $X$ may be endowed by a metric d such that $X$ has a limit point in $(X,d)$

This is an exercise I've been dealing with for a few days; I was wondering if anyone could help me with a hint or just telling me the answer. Regards
0
votes
1answer
20 views

Redefine a discrete compact set

I need to find twice continuously differentiable functions $g_i: \mathbb{R}^2 \rightarrow \mathbb{R}$ $i=1,\ldots,I$ such that the set $\{ 0,1\}^2=\{(0,0), (1,0), (0,1), (1,1) \}$ can be written as ...
0
votes
1answer
22 views

Define a compact and convex set through inequality constraints

I need to find twice continuously differentiable functions $g_i: \mathbb{R}^2 \rightarrow \mathbb{R}$ $i=1,...,I$ such that the set $\{ 0,1\}^2=\{(0,0), (1,0), (0,1), (1,1) \}$ can be written as $\{x ...
0
votes
0answers
33 views

is this space dense?

we know that $C(\partial\mathbb{D})$ as the continious functions on $\partial\mathbb{D}$ is dense in $L^2(\partial\mathbb{D})$. is it true that for every n the set $\{{f(t)e^{int}: f\in ...
0
votes
0answers
37 views

Onto continuous function on a compact metric space is isometry. [duplicate]

Let $K$ be a compact metric space with metric $d$ and suppose $f:K\rightarrow K$ is continuous and surjective (onto), and satisfies $d(f(x),f(y))\leq d(x,y),\,\forall x,y\in K$. How can we prove that ...
2
votes
1answer
32 views

Example where $f$ is discontinuous

Let $X,Y$ be topological spaces and $f: X \to Y$. I know that if $X,Y$ are not necessarily first countable (=countable nbhood base) then ''For all sequences $x_n\to x$ in $X$ it's true that $f(x_n) ...
1
vote
1answer
37 views

Proving the distance function $|f - g|_u = \sup \{ |f(x) - g(x)|: x \in S \}$ defines a metric space

Let $S$ be a closed and bounded subset of $\mathbb{R}$. Define the "functional distance" between $f$ and $g$, both functions from $S$ to $\mathbb{R}$, to be $$ |f - g|_u = \sup \{ |f(x) - g(x)|: x ...
1
vote
1answer
17 views

Is Cantor-Bendixson theorem right for a general second countable space?

The Cantor-Bendixson theorem states that If $X$ is Polish then any closed subset of $X$ can be written as the disjoint union of a perfect subset and an at most countable subset. It seems that we ...
-1
votes
2answers
25 views

Question about open and closed sets

Let $A \subseteq \mathbb{R}^n$ be an open set. does $A$ always contain a closed box?
1
vote
1answer
25 views

Extend projection on $L^2$ to one on $L^p$

if we have a closed subspace of $L^p$ called $X \cong l^2$ where the topologies of $L^p$ and $L^2$ coincide (we assume $p>2$). Then we can regard $X$ as a subspace of $L^2$, which means that he is ...
0
votes
1answer
39 views

Why is the topology of convergence in measure equivalent to this metric here?

I am currently struggeling with the topic of convergence in measure topologies. Now I read that on the space of measurable function $L^0$ on $[0,1]$ with the Borel sigma algebra and the lebesgue ...
0
votes
1answer
35 views

what are closed sets in $L^{1}(\mathbb R)$?

Consider, $L^{1}(\mathbb R)$= The space of Lebesgue integrable functions on $\mathbb R$; for $f\in L^{1}(\mathbb R),$ we define its norm, by $\|f\|_{L^{1}}=\int_{\mathbb R}|f(x)| dx$; It is well-known ...
0
votes
1answer
14 views

Finding the boundary of a set

Let $I$ be any interval in the real line and consider the set $A = I \cap \mathbb{Q} $.Notice $A \subseteq \mathbb{Q} $. then $A$ must have measure zero since $\mathbb{Q}$ does. Is the boundary of $A$ ...
0
votes
0answers
92 views

understand proof of compactness in product topology

I am trying to understand the following reasoning. Call $\mathcal{F_\lambda}$ the set of functions $a:\mathbb{N} \to \mathbb{R}$ for which $Na(i) := \sum_{j \in \mathbb{N}} n_{ij} a(j)\leq \lambda ...
0
votes
1answer
58 views

Relationship between $\|\cdot\|_p$ and $\|\cdot\|_{\infty}$

Is a knowledge fact that in $\mathbb{R}^n$ we have $$ \|\cdot\|_{\infty}=\lim_{p\to \infty}\|\cdot\|_{p} $$ where we define $$ \|x\|_{p}=\left(\sum_{i=1}^{n}x_i^p\right)^{\frac{1}{p}} $$ and $$ ...
1
vote
1answer
42 views

If $(X,d)$ is a metric space then I want to show that limit point compactness and sequential compactness are equivalent.

If $(X,d)$ is a metric space then I want to show that limit point compactness and sequential compactness are equivalent. I think I get the idea here, but had trouble writing a decent proof. Suppose ...
0
votes
2answers
29 views

set of limit points of dense set

Let $A \subseteq \mathbb{R}^{n}$ and both $A$ and $A^{c}$ are dense in $\mathbb{R}^{n}$. Can we find $x \in A$ such that $x$ is NOT a limit point of $A$? Let $B \subseteq \mathbb{R}^{n}$ and $B$ is ...
2
votes
1answer
51 views

the continuous functions with norm

I'm having trouble trying to understand what does means the first expression in particular the last term in it should we add $\|f\|_{\infty} \leq \infty$ or what i can't see what is his role ...
1
vote
1answer
23 views

Conditions for convergences of a net

I got stuck on this problem and got no clue to solve it. Can anyone one here help me? I really appreciate. Let $X$ be a set and $\mathcal{A}$ the collection of all finite subsets of $X$, ...
0
votes
0answers
25 views

Some fundamental relations in topology

Are the following relations correct? $\ \{ Normed\, Vector\, Spaces\} \subset \{Topological\, Vector\, Spaces\} \subset \{Uniform \,Spaces\} \subset \{Topological\, Spaces\}$ Then $\ \{Normed\, ...
1
vote
1answer
67 views

Metrics and Continuous Functions

The following question reads as i) Given an example of infinite metric spaces $(X,d)$ and $(Y,\delta)$ such that every function $f\colon X\to Y$ is continuous. ii) Is it possible to give an ...
0
votes
0answers
20 views

Convex in $ \mathbb{R^n}$

Prove that: [A be a convexe part $(A\subseteq \mathbb{R^n})] \implies [\forall x_1,x_2,...x_n\in A ,\forall\alpha_1,\alpha_2,...\alpha_n\ge0 $ $with$ $ \ \alpha_1+\alpha_2+...+\alpha_n=1 ...
0
votes
3answers
73 views

Closed subset of compact set is compact

If S is a compact subset of R and T is a closed subset of S,then T is compact. (a) Prove this using definition of compactness. (b) Prove this using the Heine-Borel theorem. My solution: ...
0
votes
2answers
31 views

Distance between a point and a set

The problem I'm trying to solve is Prove that $d(a, B \cup C)$ is the smaller of $d(a,B)$ and $d(a,C)$ for a point $a$ and subsets $B, C$ of a metric space. So I think what I need to show is ...
0
votes
1answer
37 views

Quick topology question

I confused myself. It is a seemingly trivial question: If $U,V,B$ are sets in a topological space $X$ and $U \subset B$ is open in $B$ and $U = U \cap V$ is it true that $U \cap V$ is open in $B \cap ...
2
votes
1answer
73 views

In $\mathbb{R}^n$ how many disjoint open sets can share a boundary

I know that in $\mathbb{R}$ you can have at most $2$ disjoint open sets that share a boundary(I believe my answer to Open Sets Boundary question proves that). My question is is there a way to extend ...
3
votes
2answers
48 views

closed ball in euclidean space

In general metric spaces the closed ball is not the closure of an open ball. However, I read that in the Euclidean space with usual metric, closed ball is the closure of an open ball. I'm having ...
0
votes
1answer
55 views

The Solution to Exercise 4, page 12, of Gamelin's “Introduction to Topology”.

I'm looking at exercise 4 in page 12 of Gamelin's Introduction to Topology. The problem is stated as follows: Suppose that $F$ is a subset of the first category in a metric space $X$ and $E$ is ...
1
vote
1answer
41 views

corollary to baire category theorem

I'm studying topology with gamelin and greene's text and I came across a corollary to the baire category theorem which states that "Let (En) be a sequence of nowhere dense subsets of a complete ...
0
votes
1answer
92 views

Reference request: Partition of unity…

I was looking for some material that could help me understand a real analysis course (1st year undergraduate). My teacher treated the following topics: Partition of unity Existence of regular ...
1
vote
1answer
27 views

compact set has a countable base

Let $K \subseteq \mathbb{R}^n$ be a compact set. Then, there exists a countable set $S \subseteq K $ such that $\overline{S} = K$ My try: Notice for any $n$, the collection $U_n = \{ B( x, ...
1
vote
1answer
18 views

Question about compact sets and coverings

Let $K \subseteq \mathbb{R}^n$ be sequentially compact. Then for every $\epsilon > 0$ there exists $x_1,...,x_m \in K $ such that $K \subseteq \bigcup_{i=1}^n B(x_i, \epsilon ) $. Proof Suppose ...
2
votes
3answers
70 views

show the supremum of the distance function of a compact metric space is finite

Let $X$ be a compact topological space and $(Y,d)$ be a metric space. Show that for every pair of continuous functions $f\colon X\to Y$ and $g\colon X\to Y$, the extended real number $$ ...
0
votes
2answers
77 views

Discontinuity of the characteristic function

Let $A \subseteq \mathbb{R}^n$. Let $f(x) = \chi_A $ be the characteristic function, and put $D = \{ x : f(x) \; \; \text{is discontinuous} \} $. Then $\partial A = D $. MY try: Let $y \in D $. ...
1
vote
2answers
36 views

graph of a continuous function is closed

Let $f: \mathbb{R} \to \mathbb{R}$ be continuous. Then $G = \{ (x, f(x) ) : x \in \mathbb{R} \} $ is a closed set. My try: Suppose $(z_n) = (x_n, f(x_n) ) $ is sequence in $G$ with limit $(x,y)$. We ...
0
votes
1answer
46 views

Every Bounded set contained in a Compact set

In a general metric space, is every bounded set contained in a compact set?
0
votes
2answers
45 views

Closure of a set is the set of limit points.

Let $A \subseteq \mathbb{R}^n$. Let $S = \{ x \in \mathbb{R}^n : \exists (x_n) \subseteq A \; \; \; s.t \; \; x_n \to x \} $ $$ \text{Claim}: \; \overline{A} = S $$ Attempt $ \overline{A} \subseteq ...
4
votes
1answer
73 views

Equal boundaries

I am given that $A \subset B \subset \mathbb{R}$, $A$ is open, $B$ is closed, and that $\partial A = \partial B$. Can I prove from this that $B$ is either equal to the closure of $A$ or it is ...
2
votes
2answers
81 views

What are the things I need to know to study Topology

I am looking to learn about Riemann Surfaces but I know that beforehand I need to study certain subjects like Metric and Topological Spaces, Complex/Real Analysis and Complex Functions. Can anyone ...
1
vote
1answer
45 views

Continuous functions on a closed subset of a topological space

Let $X$ be a topological space with $Y$ a closed subspace with relative topology. If $f:Y \rightarrow Z$ is a continuous map of topological spaces, then can $f$ always be extended to be from $X$ to ...
0
votes
1answer
34 views

Is every separately continuous function on $R^2$ continuous?

My friend asked me this question a few days ago. I felt it's not right but couldn't find a single counterexample. Any comment is appreciated.