0
votes
1answer
35 views

The set of rational numbers in the interval $(0,1)$ cannot be expressed as the intersection of a countable collection of open sets

The set of rational numbers in the interval $(0,1)$ cannot be expressed as the intersection of a countable collection of open sets I found this proof on a certain web page A direct proof would be ...
2
votes
2answers
42 views

For which compact sets can the size of the finite subcover be bounded?

I've been struggling to find a solution to this problem: For which compact sets can you set an upper bound on the number of sets in a subcover of an open cover. My understanding is that I need to ...
2
votes
3answers
27 views

Analysis proof with metric spaces

Part a) of Theorem 2.27 in baby Rudin reads (roughly) as follows: Theorem. Let $X$ be a metric space and $E\subset X$. Then the closure of $E$ is closed. Proof. Let $x\in\bar{E}^c$. Then ...
0
votes
3answers
46 views

3 questions about topology on metric space

I am reading a textbook about topology on metric space. I came over the following three 'Prove or Disprove' questions. Please: 1) comment on my work on the first two questions or leave me your own ...
0
votes
1answer
25 views

Clarification on separability in Rudin

On pg 45 of Baby Rudin we have: 22. A metric space is called separable if it contains a countable dense subset. 24. Let $X$ be a metric space in which every infinite subset has a limit point. ...
1
vote
4answers
55 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
1answer
35 views

Dense subsets and lipschitz functions

Let U and V be metric spaces. Let X be a dense subset of U. Suppose that there exists a function, h: X $\rightarrow$ V, such that h is Lipschitz with constant K. Show that there exists a function f: U ...
1
vote
1answer
29 views

Uniformly boundedness and equicontinuous set of functions

Let (X, $\mathcal{O}$) be a topological space. Suppose that {f$_{\alpha}$(x)}$_{\alpha \in A}$ is a family of functions on $\mathbb{R}$ that are uniformly bounded and equicontinuous. Prove that the ...
0
votes
0answers
26 views

Limit points of a set

If $A=\{\frac{2}{m}+\frac{3}{n}:m,n\in \mathbb N\}$, then what is the derived set of $A$ in $\mathbb R$? Definitely $0$ is a limit point of $A$. I think for all $n\in \mathbb N$, all numbers of the ...
0
votes
1answer
17 views

Limit of the supremum of a continuous function over a varying set

On a book I'm reading the author asserts that $$\lim_{\epsilon \to 0} \max_{{\bar{Q}}_{T-\epsilon}} u = \max_{{\bar{Q}}_T} u$$ where ${\bar{Q}}_{T-\epsilon}$ is the closure of a varying ...
3
votes
3answers
43 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 ...
2
votes
1answer
28 views

Closure and subbasis

Let $X$ be a topological space and $A \subset X$ with a subbasis $S$. Does it then hold that $x \in \overline{A}: \Leftrightarrow \forall s \in S: (x \in s \Rightarrow s \cap A \neq \emptyset).$ This ...
-2
votes
1answer
28 views

On Pseudometric

How a pseudometrics induces topology? Can anyone discuss on this topic or give any good reference?
1
vote
3answers
48 views

What's the closure of $(a,b)$ in discrete topology on the real number $\mathbb{R}$

In my opinion, by definition, the closure of $(a,b)$ in discrete topology on the real number $\mathbb{R}$ is $(a,b)$. However, I just saw the answer for this question is $[a,b]$. Now I am not sure ...
0
votes
1answer
21 views

Determine which of the following subsets of R2 open, closed or neither

Determine which of the following subsets of $\Bbb R^2$ are open, closed or neither: \begin{align} &1.\quad A=\{(x,y)\in \Bbb R^2:x+y=1\}\\ &2. \quad A=\{(x,y)\in \Bbb R^2: x+y>1\}\\ ...
2
votes
2answers
72 views

Topology on generalized metric space and metric space

Let $X$ be a nonempty set and $d: X\times X\to R$ be a function such that for all $x,y\in X$ and all distinct $u, v\in X$ each of which is different from $x$ and $y$ (1) $ d(x,y)\geq 0$ ; (2) ...
2
votes
1answer
42 views

If ${T_n}$ is a sequence of sets that converges to the set of irrational numbers, does $\overline{T_n}$ contain an interval for some $n$?

If $\{T_n\}$ is a sequence of sets that converges to the set of irrational numbers such that $T_1 \subseteq T_2 \subseteq T_3 \subseteq \ldots$. Must $\overline{T_n}$ contain an interval for some $n$? ...
1
vote
1answer
30 views

Topology on the real line 2

"$A\subset \Bbb{R}$ is said convex if for all $x,y\in A$ and $0\leq\lambda\leq1$ then $\lambda x+(1-\lambda)y\in A$. Show that a subset $C$ of $\Bbb{R}$ is convex if, and only if, $C$ is an interval." ...
1
vote
2answers
37 views

Let $f:[a,b]\to\mathbb R$ continuous. Prove that $G=${${(x,f(x): x\in [a,b]}$} (graph of $f$) is connected

Let $f:[a,b]\to\mathbb R$ continuous. Prove that $G=${${(x,f(x): x\in [a,b]}$} (graph of $f$) is connected Suppose $G$ is disconnected then $\exists A,B$ relatively open disjoint sets so that $A\neq ...
-1
votes
0answers
18 views

Topology on the real line [duplicate]

True or false: (Justify) If $A\subset\Bbb{R}$ is open then $A$ is an finite or contable union of open intervals.
1
vote
3answers
43 views

Equivalent alternative to delta-epsilon formulation of limit?

Is $\lim_{n \to \infty} x_n = L$ same as: $\forall n \in \mathbb N, \exists \ \varepsilon_n > 0 $ so that: $$ \ |x_n - L| \leq \varepsilon_n$$ and $$\varepsilon_n \to 0, \ ...
10
votes
1answer
106 views

Is there always an equivalent metric which is not complete?

I have seen that completeness is not a topological property like compactness or connectedness. I have seen some examples also showing that there are two equivalent metrics one of which is complete and ...
0
votes
2answers
26 views

The boundary of the union of two sets is a subset of the union of boundaries

I'm stuck on trying to get this proof started. I want to prove that $\delta(S_1 \cup S_2)\subset \delta S_1\cup\delta S_2$, where $S$ is some set. I don't need a full proof, just a hint to get ...
0
votes
1answer
84 views

How to construct a smooth curve whose range is dense in $\mathbb R^2$?

How to construct a smooth curve $f: \mathbb R \to \mathbb R^2$ whose range is dense in $\mathbb R^2$? Space-filling curves are well-known, but they cannot be smooth. The image of a smooth ...
0
votes
0answers
28 views

Conditions any dense embedding from $(0,1]$ into $[0,1]$ must satisfy

This is a proof-verification request. Suppose that $m:(0,1]\to[0,1]$ is a dense embedding. That is, $m$ is continuous; $m$ is injective; the image $m\big((0,1]\big)$ is dense in $[0,1]$; $m$ has a ...
-1
votes
0answers
46 views

Cantor-set construction problem [closed]

I'm having some trouble getting started with this proof. Any advice would be helpful. Thanks. Show that the set of numbers in the interval [0,1] having decimal expansions using only odd digits is ...
2
votes
2answers
38 views

The set of numbers of the form $k/5^n$ is dense in the real line [closed]

Show that the set of numbers of the form $k/ 5^n$, where $k$ is an integer and $n$ is a positive integer, is dense in the line. I'm having a little trouble getting started on this problem. Any ...
0
votes
3answers
76 views
+50

The set of all segments is convex

If $X,Y\subset \mathbb R^n$ are convex sets, is it true the union of all segments $[x,y]$, where $x\in X$ and $y\in Y$ is a convex set? I've drawn pictures and I convinced myself that this is true, ...
0
votes
2answers
24 views

Sequential Equivalence Implies Topological Equivalence

Define two metric spaces $(M,d)$ and $(M,\rho)$ to be equivalent, denoted $d\sim p$, to mean that: Topological Definition $\forall x\in M: \forall \epsilon>0 \exists \delta_1>0, \delta_2>0: ...
0
votes
1answer
28 views

Partition of Unity's Lemma

Let $V\subset\mathbb{R}^n$ compact, $\Omega\subset\mathbb{R}^n$ open, $V\subset\Omega$, $\delta:=\inf\{|x-y|\mid x\in V,y\notin\Omega\}$, $U:=\left\{x \mid |x-y|<\frac{\delta}{2}\,\,\text{for ...
2
votes
1answer
37 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
47 views

P-norm Unit Ball

Proof that for $0<p<1$, $p\in \Bbb{R}$ $$\|(x,y)\|_p=(|x|^p+|y|^p)^{\frac{1}{p}}$$ doesn't define a norm in $\Bbb{R}^2$. However, $$d_p((x_1,x_2),(y_1,y_2))=\sum_{i=1}^2|x_i-y_i|^p$$ defines a ...
1
vote
1answer
31 views

Prove that for $0<p<1$, $|x-y|^p$ is a metric space on $\mathbb R^{n}$

Define the function $f_p : \mathbb R^{n} \to \mathbb R^{n}$ for $n ≥ 2$ by $f_p(x) = \sum_{k=1}^{n} \lvert x\rvert^{p}$. Show that for $0<p<1$, we get $d_b(x,y) = f_p(x-y)$ is a metric on ...
0
votes
2answers
36 views

Proving that a set $A$ is dense in $M$ iff $A^c$ has empty interior

Prove that a set $A$ is dense in a metric space $(M,d)$ iff $A^c$ has empty interior. Attempt: I think I proved the converse correctly, but I'm not sure how to start the forward direction. ...
2
votes
1answer
46 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!
0
votes
1answer
18 views

Question about pointwise convergence of sequences in the box and product topologies.

Can someone please verify my proof or offer suggestions for improvement? I'm aware that there may be answers floating elsewhere, but I need help with my proof in particular. $\textbf{Note:}$ This is ...
0
votes
2answers
43 views

For a compact set $K\subset \Bbb R^n $ prove the following :

For a compact set $K\subset \Bbb R^n $ and $\delta>0$ show that that there exists a finite number of elements in $K$, say $x_1,x_2,\dots,x_k$ such that any other element $x$ of $K$ is at a ...
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 ...
3
votes
0answers
14 views

Show that if $X_\alpha$ is Hausdorff for all $\alpha$, then $\prod X_\alpha$ is Hausdorff under the box and product topologies.

Can someone please verify my proof? I am aware that there is a similar question posted elsewhere, but I need help with my proof in particular. $\textbf{Note:}$ This is not homework. Show that if ...
0
votes
3answers
86 views

Are all the points in a nonempty open set limit points?

My conjecture is that given any open set $A\subseteq\mathbb{R}$, all points $a\in A$ are limit points. Prove, or if untrue, disprove by constructing a counterexample. A few definitions for ...
2
votes
1answer
32 views

Doubts about definition of open sets in “Understanding Analysis” by Stephen Abbott

In the book "Understanding Analysis" by Stephen Abbott, the author defines an open set as: A set $O \subseteq \mathbb{R}$ is open if for all points $a \in O$ there exists an ...
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
0
votes
1answer
33 views

What does it mean “sequence with infinite range”

I'm trying to understand this phrase Find a sequence with infinite range that converges only to $0$. What does it mean "sequence with infinite range"? Thanks
0
votes
0answers
26 views

Show that if $F$ is continuous, then it is continuous in each variable separately.

Can someone please verify my proof? I am aware that there may be a similar question posted elsewhere, but I need help with my proof in particular. $\textbf{Note:}$ This is not homework. Let $F: X ...
0
votes
0answers
20 views

Define the 4 types of interval subsets of the real numbers.

Define the 4 types of interval subsets of the real numbers. Is the union of an arbitrary number of open intervals also an open interval? Is the intersection of an arbitrary number of open intervals ...
4
votes
0answers
64 views

Equivalence of Lebesgue Measurablity

Hello Mathematics Community. I am having some difficulties with the following problem dealing with Lebesgue Measure and its equivalent interpretation. I will first include the definitions which I am ...
3
votes
0answers
57 views

Compactness and Lipschitz functions

I am very stumped by this question: Suppose (K, d) is a compact metric space. Let f be any function, f: K $\rightarrow \mathbb{C}$, not necessarily continuous. Prove that for any $\epsilon > ...
1
vote
0answers
29 views

Showing a subset of $\;\Bbb R^2\;$ cannot be the set of limit points of any other set

I will appreciate any insight in the following proof (if, indeed, it is a proof): Let $$F:=\left\{\;(x,y)\in\Bbb R^2\;;\;\;xy\in\Bbb Q\;\right\}$$ Prove that there doesn't exist $\;A\subset\Bbb ...
0
votes
2answers
40 views

Product metric spaces is again a metric space

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, and let: $$ d_2 ((x_1,y_1),(x_2,y_2)) = \left[d_X(x_1,x_2)^2 + d_Y (y_1,y_2)^2 \right]^{\frac{1}{2}} $$ for the points $(x_1,y_1)$ and $(x_2,y_2)$ in $X ...
1
vote
1answer
29 views

Mapping on induced topology and distance metric

Let $(X, d)$ be a metric space. Let $τ$ be the metric topology on $X$ induced by $d$. For $A ⊆ X$ , let $d(x, A) := \inf_{a∈A} d(x, a) $ for $x ∈ X$ (a) If $f (x) := d(x, A)$ (for a fixed subset ...