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Topology glossary

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no clear distinction between different areas of topology, this glossary focuses primarily on general topology and definitions that are fundamental to a broad range of areas. See the article on topological spaces for basic definitions and examples, and see the article on topology for a brief history and description of the subject area.

The following articles may also be useful. These either contain specialised vocabulary within general topology or provide more detailed expositions of the definitions given below. The list of general topology topics will also be very helpful.

All spaces in this glossary are assumed to be topological spaces unless stated otherwise.

Isotonicity: Every set is contained in its closure.
  • Idempotence: The closure of the closure of a set is equal to the closure of that set.
  • Preservation of binary unions: The closure of the union of two sets is the union of their closures.
  • Preservation of nullary unions: The closure of the empty set is empty.

  • d(x, y) ≥ 0
  • d(x, x) = 0
  • if   d(x, y) = 0   then   x = y     (identity of indiscernibles)
  • d(x, y) = d(y, x)     (symmetry)
  • d(x, z) ≤ d(x, y) + d(y, z)     (triangle inequality)

  • The function d is called a metric on M.

    The empty set and X are in T.
  • The union of any collection of sets in T is also in T.
  • The intersection of any pair of sets in T is also in T.

  • The collection T is called a topology on X.

    if U is in Φ, then U contains { (x, x) : x in X }.
  • if U is in Φ, then { (y, x) : (x, y) in U } is also in Φ
  • if U is in Φ and V is a subset of X × X which contains U, then V is in Φ
  • if U and V are in Φ, then UV is in Φ
  • if U is in Φ, then there exists V in Φ such that, whenever (x, y) and (y, z) are in V, then (x, z) is in U.

  • The elements of Φ are called entourages, and Φ itself is called a uniform structure on U.