User:Leigh.Samphier/Topology

Topology
User:Leigh.Samphier/Topology/Definition:Connected Manifold

User:Leigh.Samphier/Topology/Definition:Connected Manifold/Definition 2

User:Leigh.Samphier/Topology/Equivalence of Definitions of Connected Manifold


 * Definition:Connected Manifold/Definition 1


 * Topological Manifold is Locally Connected


 * Topological Manifold is Locally Path-Connected


 * Topological Manifold is Locally Compact


 * Locally Euclidean Space is Locally Connected


 * Open Ball is Connected


 * Closed Ball is Connected


 * Locally Euclidean Space is Locally Path-Connected


 * Open Ball is Path-Connected


 * Closed Ball is Path-Connected


 * Convex Set is Path-Connected


 * Locally Euclidean Space is Locally Compact


 * Locally Euclidean Space is Locally Compact/Proof 1


 * Locally Euclidean Space is Locally Compact/Proof 2


 * Locally Euclidean Space has Countable Neighborhood Basis Homeomorphic to Closed Balls


 * Locally Euclidean Space is First-Countable


 * Locally Euclidean Space has Countable Local Basis Homeomorphic to Open Balls


 * Null Sequence induces Neighborhood Basis of Closed Sets in Metric Space


 * Closed Ball in Metric Space is Closed Neighborhood


 * Homeomorphic Image of Neighborhood Basis is Neighborhood Basis


 * Homeomorphic Image of Local Basis is Local Basis


 * Local Basis of Open Subspace iff Local Basis


 * Neighborhood Basis of Open Subspace iff Neighborhood Basis


 * Neighborhood in Open Subspace


 * Sequence Lemma


 * Countable Product of Sequentially Compact Spaces is Sequentially Compact


 * Sequence Characterization of Open Sets


 * Final Topology with respect to Mapping