# Tagged Questions

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### Paradox in connection with definition of limit points and order limit theorem?

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I come across something that appears (to me) as a paradox. Let me first write down one definition and two theorems that ...
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### Exercise 3.3.8 from Understanding Analysis by Stephen Abbott

Motivation: trying to prove that if $K \subseteq \mathbb{R}$ is compact (and thus, by the Heine-Borel theorem, closed and bounded), then this implies that any open cover for $K$ has a finite subcover. ...
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### The boundary of an open subset of $[0,1]$ containing all rationals in $(0,1)$

If $A\subset [0,1]$ is the union of open intervals $(a_i,b_i)$ such that each rational number of $(0,1)$ is contained in some $(a_i,b_i)$, prove that the boundary (frontier) of $A$ is $[0,1]-A$. ...
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### About the proof of the Heine-Borel Theorem

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I have a question about the prove of theorem 3.3.4 on page 84 (i.e. the Heine-Borel theorem). To be more specific, let us ...
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### A subset of $[0,1]\times[0,1]$ containing at most one point from each horizontal and vertical section whose boundary is $[0,1]\times[0,1]$

How can one build a subset $A\subset [0,1]\times[0,1]$ containing at the most one point from each horizontal and each vertical section and whose boundary (frontier) is $[0,1]\times[0,1]$? I don't ...
I only have studied basic topology, which means i haven't studied any about differentiable manifolds. I just skimmed pages on wikipedia. Here is a simple illustration on a basic situation. Let $E$ ...