Reference cards

DIFFERENTIAL GEOMETRY

Objects

• Atlasesa set of charts whose domains have union the whole manifold
• Chaotic and Discrete Topologiesthe smallest and largest topologies on any set: only ∅ and M open, or every subset open
• Chart Transition Mapsthe map y ∘ x⁻¹ from x(U ∩ V) to y(U ∩ V)
• Chartsa pair (U, x) with U open and x a homeomorphism onto an open subset of ℝᵈ
• Complementthe points of a set that a given subset misses
• Connected and Disconnected Spacesa space admitting no pair of disjoint non-empty open sets that together cover it
• Continuous Mapa map whose preimages of open sets are open
• Homeomorphisma bijection continuous in both directions, so the open sets correspond exactly
• Injective, Surjective, Bijectivethe two conditions that together let a map be turned around
• Open and Closed Setsopen means the subset belongs to the topology; closed means its complement does
• Open Covers and Compact Spacesevery open cover has a finite subcover
• Powersetall the subsets of a set, collected into one set
• Preimagename a subset of the target, get back every point of the domain sent into it
• Standard Topologythe subsets of ℝᵈ in which every point has a soft ball about it lying inside the set
• Subset Topologyintersect every open set of the ambient space with the subset
• Topological Manifolda space in which every point has an open neighbourhood homeomorphic to an open subset of ℝᵈ
• Topology and Topological Spacea family of subsets holding ∅ and M, closed under pairwise intersection and arbitrary union