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# 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. ## Resources - [Topology on Wikipedia](https://en.wikipedia.org/wiki/Topology) - [Topology and Geometry by Glen E. Bredon](https://www.springer.com/gp/book/9780387909213) - [Topology by James Munkres](https://www.pearson.com/us/higher-education/program/Munkres-Topology-2nd-Edition/PGM119237.html)