Topology Cheatsheet
Basic Concepts
- Topology studies properties of space that are preserved under continuous transformations such as stretching and bending.
- A topological space is a set with a collection of open sets that satisfy certain axioms.
- A subset of a topological space is called closed if its complement is open.
Topological Spaces
- A topological space is a set X with a collection T of subsets of X called open sets.
- The empty set and X are both open sets.
- The intersection of any finite number of open sets is an open set.
- The union of any number of open sets is an open set.
Continuous Functions
- A function between two topological spaces is continuous if the preimage of every open set is open.
- A homeomorphism is a bijective continuous function with a continuous inverse.
Compactness
- A topological space is compact if every open cover has a finite subcover.
- A subset of a topological space is compact if it is compact as a topological space with the subspace topology.
Connectedness
- A topological space is connected if it cannot be expressed as the union of two non-empty disjoint open sets.
- A subset of a topological space is connected if it is connected as a topological space with the subspace topology.
Manifolds
- A topological manifold is a topological space that is locally Euclidean.
- A differentiable manifold is a manifold with a differentiable structure.
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