Proof of Topological Spaces
Recall the definition of a topological space.
A topology on a set X is a collection \mathcal{T} of subsets of X having the following properties:
The empty set \varnothing and the set itself X are in \mathcal{T}.
Any union of elements in \mathcal{T} is in \mathcal{T}.
Any finite intersection of elements in \mathcal{T} is in \mathcal{T}.
A topological space is an ordered pair (X,\mathcal{T}) such that X is a set and \mathcal{T} is a topology on X.
We prove the following are examples of topological spaces.
Let X be a finite set, and let \mathcal{T} = \mathcal{P}(X).
Then (X,\mathcal{T}) is the Discrete Topology (Finite). (S000001)
We must show the space satisfies the three requirements to be a topology
(1) The empty set \varnothing and the set itself X are in \mathcal{T}.
\qquad By definition of a powerset, \varnothing \in \mathcal{P}(X), and X \in \mathcal{P}(X).
(2) Any union of elements in \mathcal{T} is in \mathcal{T}.
\qquad By THEOREM of powerset, \mathcal{P}(X) is closed under set union. \qquad By THEOREM of powerset, \mathcal{P}(X) is closed under set union.
(3) Any finite intersection of elements in \mathcal{T} is in \mathcal{T}.
\qquad By THEOREM of powerset, \mathcal{P}(X) is closed under set intersection.
Let X be a countably infinite set, and let \mathcal{T} = \mathcal{P}(X).
Then (X,\mathcal{T}) is the Discrete Topology (Countable). (S000002)
We must show the space satisfies the three requirements to be a topology
(1) The empty set \varnothing and the set itself X are in \mathcal{T}.
\qquad By definition of a powerset, \varnothing \in \mathcal{P}(X), and X \in \mathcal{P}(X).
(2) Any union of elements in \mathcal{T} is in \mathcal{T}.
\qquad By THEOREM of powerset, \mathcal{P}(X) is closed under set union.
(3) Any finite intersection of elements in \mathcal{T} is in \mathcal{T}.
\qquad By THEOREM of powerset, \mathcal{P}(X) is closed under set intersection.
Let X be an uncountably infinite set, and let \mathcal{T} = \mathcal{P}(X).
Then (X,\mathcal{T}) is the Discrete Topology (Uncountable). (S000003)
We must show the space satisfies the three requirements to be a topology
(1) The empty set \varnothing and the set itself X are in \mathcal{T}.
\qquad By definition of a powerset, \varnothing \in \mathcal{P}(X), and X \in \mathcal{P}(X).
(2) Any union of elements in \mathcal{T} is in \mathcal{T}.
\qquad By THEOREM of powerset, \mathcal{P}(X) is closed under set union.
(3) Any finite intersection of elements in \mathcal{T} is in \mathcal{T}.
\qquad By THEOREM of powerset, \mathcal{P}(X) is closed under set intersection.
(TO-DO: Foundations, prove power set properties, link here.)
Let X be a countably infinite set, and let \mathcal{T} be all sets U such that X \backslash U is finite, and the empty set.
Then (X,\mathcal{T}) is the Cofinite Topology (Countable). (S000015)
We must show the space satisfies the three requirements to be a topology
(1) The empty set \varnothing and the set itself X are in \mathcal{T}.
\qquad The empty set is open by definition. The set X \backslash X is the empty set, which is finite, thus X is open.
(2) Any union of elements in \mathcal{T} is in \mathcal{T}.
\qquad All open sets are either infinite or empty, thus the intersection of two infinite sets (or an infinite and empty set) is infinite, and thus in the topology.
(3) Any finite intersection of elements in \mathcal{T} is in \mathcal{T}.
\qquad The intersection of two open sets in \mathcal{T} is always infinite, as if it were finite A \backslash B would be infinite, and thus X \backslash B would be infinite, and B would not be open. Repeat induction.
(TO-DO: De Morgans laws, better proofs)