Date: 2024-03-24 | Article ID: 93875 |

A **topological space** is one with enough structure for defining continuity. A **topological manifold** is intermediate between a topological space and a smooth manifold. A **smooth manifold** is one with enough structure for defining derivatives. The smooth manifold R^{N}, the set of N-tuples of real numbers equipped with the standard smooth structure, is a familiar example from which all others can be constructed patchwise. This article is a brief reminder of the basic ideas. A list of relationships between various types of topological spaces is given at the end, summarized graphically by a Venn diagram.

**2024-03-24**(added sections reviewing the concept of a CW complex, added a section about other generalizations)**2024-02-25**(replaced a sentence in the abstract with a more informative one, added references about topological manifolds, added a section relating metric spaces to manifolds, refined the definition of smooth structure to be maximal smooth atlas, added more references in the section about diffeomorphisms, clarified the dimensions in the section about diffeomorphisms, removed a ill-conceived reason in the section about diffeomorphisms, added a clarification in the section about diffeomorphisms, added new sections about relationships between different special kinds of topological spaces)**2023-11-12**(replaced Euclidean with euclidean)**2023-10-01**(separated two facts that were incorrectly conflated in the section about diffeomorphisms)**2022-06-11**(updated references to other articles in this series to link to html abstract pages instead of to pdfs. Didn't change the version number/date)**2022-02-05**(reformatted title, fixed pdf metadata)**2022-01-16**(first version)

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