 
Summary: Definition MunkTop.12.1: T is a topology on X if and only if T (X)
and T and X T and for every S T , S T and for every U, V
T , U V T .
Definition MunkTop.12.2: (X, T ) is a topological space if and only if T
is a topology on X.
Definition MunkTop.12.3: U is open in (X, T ) if and only if (X, T ) is
a topological space and U T .
Definition MunkTop.12.4.a: If (X, T ) and (X, T ) are topological spaces
then T is finer than T on X if and only if T T .
Definition MunkTop.12.4.b: If (X, T ) and (X, T ) are topological spaces
then T is strictly finer than T on X if and only if T T and T = T .
Definition MunkTop.12.4.c: If (X, T ) and (X, T ) are topological spaces
then T is coarser than T on X if and only if T T .
Definition MunkTop.12.4.d: If (X, T ) and (X, T ) are topological spaces
then T is strictly coarser than T on X if and only if T T and T = T .
Definition MunkTop.13.1: B is a basis for a topology on X if and only
if B (X) and for every x X, there exists B B such that x B and for
every x X, for every B1, B2 B, if x B1 B2 then there exists B3 B
such that x B3 and B3 B1 B2.
Definition MunkTop.13.2: If B is a basis for a topology on X then the
