## New answers tagged compactness

2
votes

### Density of subsequences in Bolzano-Weierstrass

$\newcommand{\N}{\mathbb N}\newcommand{\R}{\mathbb R}\newcommand{\md}{\ (\operatorname{mod}2)}$This is to present a formalized version of the nice answer by Christian Remling.
Suppose that a function $...

6
votes

Accepted

### Density of subsequences in Bolzano-Weierstrass

There is no such function as soon as $M$ has at least two points. Let's call them $0,1$.
Given an $f$, let $x_n=0$ for $n\le f(1)$, forcing us to take $x_{n_1}=0$. Let's say we took $n_1=1$. There are ...

0
votes

### History of limit point compact -/-> compact example

Space $\ S_\Omega,\ $ and its generalization for higher initial ordinal numbers was the very first example of this type, and it was introduced by the authors of the notion of bicompact spaces -- by P....

1
vote

### History of limit point compact -/-> compact example

The earliest reference I can find is Dugundji's book Topology, published in 1966, where this result appears as Ex. 2 on page 239.
Here's how I found that. First, this result appears in Kelley's book ...

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