The Topology glossary reference article from the English Wikipedia on 24-Jul-2004
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Topology glossary

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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 on 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. See basic set theory, axiomatic set theory, and function for definitions concerning sets and functions. 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 and list of examples in general topology will also be very helpful.

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

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Table of contents
1 A
2 B
3 C
4 D
5 E
6 F
7 G
8 H
9 I
10 K
11 L
12 M
13 N
14 O
15 P
16 Q
17 R
18 S
19 T
20 U
21 W

A

B

C

D

E

F

The empty set is not in F.
  • The intersection of any finite number of elements of F is again in F.
  • If A is in F and if B contains A, then B is in F.

  • G

    H

    I

    K

    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.
  • If c is a function from the power set of X to itself, then c is a closure operator if it satisfies the Kuratowski closure axioms. The Kuratowski closure axioms can then be used to define a topology on X by declaring the closed sets to be the fixed pointss of this operator, i.e. a set A is closed if and only if c(A) = A.

    L

    M

    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 a metric on M, and d(x, y) is the distance between x and y. The collection of all open balls of M is a base for a topology on M; this is the topology on M induced by d. Every metric space is Hausdorff and paracompact (and hence normal and Tychonoff). Every metric space is first-countable.

    N

    O

    P

    Q

    R

    S

    T

    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 a topology on X.

    U

    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.

    W