Summary: Math 446­646
Important facts about Topological Spaces
· A topology on X is a collection T of subsets of X such that:
(1) T
(2) X T
(3) If A and B are in T, then A B T.
(4) If for each I, A T, then I A T.
· The elements of T are called open sets.
· A subset F of X is closed if Fc
is open (i.e., if Fc
T).
· A subset N X is a neighborhood of a point x if there is an open set O T
such that x O N.
· Let T1 and T2 be two topologies on the same space X. If T1 T2, we say that
T1 is coarser than T2, or that T2 is finer than T1.
· Some important examples:
(1) Ttriv = {X, }, the trivial topology.
(2) Tdis = P(X), the discrete topology. In the discrete topology, every
subset of X is both open and closed.
(3) If X has infinitely many elements, TF = {A X| Ac

Source: Alfonseca-Cubero, Maria - Department of Mathematics, North Dakota State University

Collections: Mathematics