[a set/sets] [an element/elements of x]
[x is in/from y @ x is an element of y]
[x belongs/belong to y @ x is in y]
[a subset/subsets of x] [the nst of x]
[x is close to y] [x is closed]
[x is compact]

Definition. Let S be a set.
A subset of S is a set A such that every element of A is in S.

Axiom. The nst of every set is a set.

Axiom. Let A be a set and B be a subset of A.
Then the nst of B is a subset of the nst of A.

Definition. Let M be a set.
M is closed iff for every element t of every set
if some element of the nst of M is close to t
then t belongs to M.

Definition. Let M be a set.
M is compact iff every element of the nst of M
is close to some element of M.

Proposition.
Every closed subset of every compact set is compact.
