Subset of a finite set is finite - Mathematics Stack Exchange
Nov 18, 2012 · A word on the axiom of choice and finiteness.The above proof shows that finite sets are Dedekind-finite.There are other ways of defining finiteness, all which are true for finite sets, but may also be true for infinite sets.
prove that any finite set in a metric space is compact
Apr 3, 2021 · To show that it is bounded, let F be a finite set, then since it is finite, by the arch median principle of natural numbers, there exists an infimum and supremum of the set therefore you can find an M > 0 s.t for some point x contained in F, abs(x) <= M and I will leave it to you to determine the other side of this. So F is bounded.
Any finite set is compact; what exactly is a finite set?
Dec 15, 2014 · A trivial example is every topology on a finite set: all subsets of a finite set are finite, thus also all compact subsets of this finite set are finite. Another example is the discrete topology (on any set), i.e. the topology where every one-element subset is open, or (equivalently) every subset is open.
A finite set is closed - Mathematics Stack Exchange
Jun 9, 2015 · Although since the OP tag this as real analysis I assume he might be talking of any finite dimensional metric space, in which case the finite sets are always closed. $\endgroup$ – Alonso Delfín Commented Jun 8, 2015 at 21:56
elementary set theory - What do finite, infinite, countable, not ...
Clearly every finite set is countable, but also some infinite sets are countable. Note that some places define countable as infinite and the above definition. In such cases we say that finite sets are "at most countable".
real analysis - A finite set always has a maximum and a minimum ...
Set A is finite iff every non-empty family of subsets of A has a minimal element [ordered by strict inclusion ' $\subset$ ']- A. Tarski via Suppes Furthermore, if one such family of subsets, F, has a minimal element , then it must also have a maximal element via the following argument:
measure theory - Definition of a $\sigma$ - finite set
I know the definition of a $\\sigma$-finite measure. But I found a problem in which it asks to show a particular set is $\\sigma$ finite? But what is a $\\sigma$ finite set? This is the problem I fou...
Definition of $\sigma$-finite - Mathematics Stack Exchange
It could happen that some decomposition of a $\sigma$-finite set into countably many disjoint subsets ...
elementary set theory - The union of finite sets is a finite set ...
The union of two finite sets is a finite set. Let $X,Y$ be finite sets. That means that there are $n, m\in \omega$ such that $X \sim n$ and $Y \sim m$, i.e. there are ...
Why set of natural numbers is infinite, while each natural number …
In his book Analysis Vol. 1, author Terence Tao argues that while each natural number is finite, the set of natural numbers is infinite (though has not defined what infinite means yet). Using Peano...