# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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!
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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
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### 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
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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 $\;A\subset\Bbb ... 2answers 40 views ### Product metric spaces is again a metric space Let$(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 ...
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 ...