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Math 446646 Important facts about Topological Spaces

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
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