Download MATH 600 Topology Cheatsheet and more Slides Algebra in PDF only on Docsity! MATH 600 Topology Cheatsheet This cheatsheet is not, of course, complete. A topological space is a pair (X,O), where X is a set and O is a collection of open subsets of X so that: 1. X ∈ O,∅ ∈ O 2. For any collection {Oα}α∈Λ of open sets, ⋃ α∈Λ Oα is an open set. 3. For any finite collection O1, . . . , ON of open sets, N⋂ i=1 Oi is an open set. A base (or basis) for the toplogical space (X,O) is a collection B ⊆ O such that any O ∈ O has O = ⋃ α∈Λ Bα for some {Bα}α∈Λ ⊂ B. An open cover for X is a collection of open sets {Oα}α∈Λ with X = ⋃ α∈Λ Oα. A topological space (X,O) is: • Hausdorff if for every pair of points p, q ∈ X there are open sets U, V ∈ O with p ∈ U, q ∈ V and U ∩ V = ∅. • second-countable if it has a countable base. • connected if there are not two disjoint, nonempty, open subsets which cover X • path-connected if every pair of points can be joined by a path. • locally path-connected if there is a base of path-connected sets. • compact if every open cover {Oα}α∈Λ has a finite subcover, i.e. there are α1, . . . , αN ∈ Λ for which X = N⋃ i=1 Oαi . • locally compact if every point p ∈ X lies in an open set O, which is contained in a compact set K. Given a topological space (X,O) and a subset S ⊆ X, we can form a topological space (S,O∩S) by setting O ∩ S = {O ∩ S|O ∈ O}. We say that S has the subspace topology from (X,O), and call (S,O ∩ S) a topological subspace of (X,O). (As an exercise, prove that a topological subspace is a topological space in its own right.) Given a topological space (X̃,O) and an equivalence relation ∼ on X̃, let X = X̃/∼ = {[x]|x ∈ X}. We can form a topological space (X,O/∼) by setting O/∼ = {U ⊆ X|{x|[x] ∈ U} ∈ O}. We call (X,O/∼) the quotient of X by ∼. Given topological spaces (X,OX) and (Y,OY ), we can form a topological space (X×Y,OX×OY ) by setting OX × OY to be the set of all unions of products OX × OY , where OX ∈ OX and OY ∈ OY . We call (X × Y,OX ×OY ) the product of (X,OX) and (Y,OY ). (Note: when the number of factors in the product is not finite, there are some subtleties as to how to gen- eralise this definition! This won’t concern us, however.) 1
