See also Topology on ProvenMath.
(X,G) is a topological space if and only if the following conditions hold. 1. X, G are sets and G c P(X) 2. O :- G and X :- G 3. /\(A,B:-G) AnB :- G 4. /\(McG) u(M) :- GOpen set - definition
Let (X,G) be a topological space. We define that A is an open set if and only if A :- G.Closed set - definition
Let (X,G) be a topological space. We define that A is a closed set if and only if A c X and X\A :- G.Interior - definition
Let (X,G) be a topological space and let A c X.
We define that int(A) = u({U : U c A and U is open}).
Closure - definition
Let (X,G) be a topological space and let A c X.
We define that clo(A) = n({F : A c F and F is closed}).
Math jargon
The collection of open sets contains the empty set and the whole space.
The intersection of two open sets is an open set.
By induction, the intersection of a finite number of open sets is an open set.
The union of any number of open sets is an open set.
Since closed sets are complements of open sets,
by De Morgan's laws for collections of sets we have what follows.
The collection of closed sets contains the empty set and the whole space.
The union of two closed sets is a closed set.
By induction, the union of a finite number of closed sets is a closed set.
The intersection of any number of closed sets is a closed set.
The interior of A is the union of all open sets contained in A.
Hence it is also contained in A and it is an open set.
Consequently, the interior of A is the largest open set contained in A.
The closure of A is the intersection of all closed sets which contain A.
Hence it also contains A and it is a closed set.
Consequently, the closure of A is the smallest closed set which contains A.
If (X,G) is a topological space and A, B c X then we have the following:
(1) int(A) c A c clo(A) c X
(2) A is open <=> A = int(A)
(3) A is closed <=> A = clo(A)
(4) A c B => int(A) c int(B)
(5) A c B => clo(A) c clo(B)
(6) int(X\A) = X \ clo(A)
(7) clo(X\A) = X \ int(A)
(8) int(A n B) = int(A) n int(B)
(9) clo(A u B) = clo(A) u clo(B).
Proof
It is easily seen from the definitions that (1), (2), and (3) hold.
(4)
Suppose that A c B. Then int(A) c A c B.
Now, int(A) is an open set contained in B.
Since int(B) is the largest open set contained in B,
it follows that int(A) c int(B).
(5)
Suppose that A c B. Then A c B c clo(B).
Now, clo(B) is a closed set which contains A.
Since clo(A) is the smallest closed set containing A,
it follows that clo(A) c clo(B).
(6)
Follow the calculations below.
clo(A) =
n({F : A c F and F is closed}) =
n({F : X\F c X\A and X\F is open}) =
n({X\U : U c X\A and U is open}) =
X \ u({U : U c X\A and U is open}) =
X \ int(X\A).
Hence int(X\A) = X \ clo(A).
(7)
By (6), X \ clo(X\A) = int(X\(X\A)) = int(A).
Hence clo(X\A) = X \ int(A).
(8)
We have int(A) c A and int(B) c B.
Hence int(A) n int(B) c A n B.
The set int(A) n int(B) is open because it is the intersection of two open sets.
Since int(A n B) is the largest open set contained in A n B,
it follows that int(A) n int(B) c int(A n B).
Now, since A n B c A and A n B c B,
it follows that int(A n B) c int(A) and int(A n B) c int(B).
Hence int(A n B) c int(A) n int(B).
We have showed that int(A n B) = int(A) n int(B).
(9)
We have A c clo(A) and B c clo(B).
Hence A u B c clo(A) u clo(B).
The set clo(A) u clo(B) is closed because it is the union of two closed sets.
Since clo(A u B) is the smallest closed set which contains A u B,
it follows that clo(A u B) c clo(A) u clo(B).
Now, since A c A u B and B c A u B,
it follows that clo(A) c clo(A u B) and clo(B) c clo(A u B).
Hence clo(A) u clo(B) c clo(A u B).
We have showed that clo(A u B) = clo(A) u clo(B).