Munkres’ Proof of Well Ordering Property
In Munkres’ Topology the proof for the Well-Ordering Property is stated as follows: I’m having confusion with the first and second underlined part: “Let A be the set of all positive integers n for …
Tag: elementary-set-theory
Using Zorn’s lemma to show a function maps part of a set one to one onto another set
Let $A$ and $B$ be any two sets. Let $X$ be the collection of functions that map a subset of $A$ one to one onto a subset of $B$. Define a partial order on $X$ and use Zorn’s lemma to show that there …
Show that $[2,infty)$ is also uncountable
Question: Assume $A$ is the only uncountable set that we currently know of, where $A= { xin (0,1): x text{ is a decimal fraction consisting only of combinations of 0 and 1} }$ From …
Proof that a sequence of set has a set dense somewhere in $[a,b]$
Is the following proof correct? Proposition: if we have a sequence of set $U_i$ such as $bigcup_{iin mathbb{N}} U_i=[a,b]$ then there exist a $i$ such as $U_i$ is dense somewhere in $[a,b]$ …