Normal space
topological spaces | |
---|---|
Kolmogorov classification | |
T0 | (Kolmogorov) |
T1 | (Fréchet) |
T2 | (Hausdorff) |
T2½ | (Urysohn) |
completely T2 | (completely Hausdorff) |
T3 | (regular Hausdorff) |
T3½ | (Tychonoff) |
T4 | (normal Hausdorff) |
T5 | (completely normal Hausdorff) |
T6 | (perfectly normal Hausdorff) |
In
Definitions
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A T4 space is a T1 space X that is normal; this is equivalent to X being normal and Hausdorff.
A completely normal space, or hereditarily normal space, is a topological space X such that every
A T5 space, or completely T4 space, is a completely normal T1 space X, which implies that X is Hausdorff; equivalently, every subspace of X must be a T4 space.
A perfectly normal space is a topological space in which every two disjoint closed sets and can be
A T6 space, or perfectly T4 space, is a perfectly normal Hausdorff space.
Note that the terms "normal space" and "T4" and derived concepts occasionally have a different meaning. (Nonetheless, "T5" always means the same as "completely T4", whatever that may be.) The definitions given here are the ones usually used today. For more on this issue, see History of the separation axioms.
Terms like "normal regular space" and "normal Hausdorff space" also turn up in the literature—they simply mean that the space both is normal and satisfies the other condition mentioned. In particular, a normal Hausdorff space is the same thing as a T4 space. Given the historical confusion of the meaning of the terms, verbal descriptions when applicable are helpful, that is, "normal Hausdorff" instead of "T4", or "completely normal Hausdorff" instead of "T5".
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Examples of normal spaces
Most spaces encountered in mathematical analysis are normal Hausdorff spaces, or at least normal regular spaces:
- All metric spaces (and hence all metrizable spaces) are perfectly normal Hausdorff;
- All pseudometrisable spaces) are perfectly normal regular, although not in general Hausdorff;
- All compact Hausdorff spaces are normal;
- In particular, the Stone–Čech compactification of a Tychonoff space is normal Hausdorff;
- Generalizing the above examples, all paracompactHausdorff spaces are normal, and all paracompact regular spaces are normal;
- All paracompact topological manifolds are perfectly normal Hausdorff. However, there exist non-paracompact manifolds that are not even normal.
- All totally ordered setsare hereditarily normal and Hausdorff.
- Every regular second-countable space is completely normal, and every regular Lindelöf space is normal.
Also, all
Examples of non-normal spaces
An important example of a non-normal topology is given by the Zariski topology on an algebraic variety or on the spectrum of a ring, which is used in algebraic geometry.
A non-normal space of some relevance to analysis is the
Properties
Every closed subset of a normal space is normal. The continuous and closed image of a normal space is normal.[6]
The main significance of normal spaces lies in the fact that they admit "enough"
, as expressed by the following theorems valid for any normal space X.Urysohn's lemma: If A and B are two disjoint closed subsets of X, then there exists a continuous function f from X to the real line R such that f(x) = 0 for all x in A and f(x) = 1 for all x in B. In fact, we can take the values of f to be entirely within the
More generally, the Tietze extension theorem: If A is a closed subset of X and f is a continuous function from A to R, then there exists a continuous function F: X → R that extends f in the sense that F(x) = f(x) for all x in A.
The map has the lifting property with respect to a map from a certain finite topological space with five points (two open and three closed) to the space with one open and two closed points.[7]
If U is a locally finite
In fact, any space that satisfies any one of these three conditions must be normal.
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Relationships to other separation axioms
If a normal space is
A topological space is said to be pseudonormal if given two disjoint closed sets in it, one of which is countable, there are disjoint open sets containing them. Every normal space is pseudonormal, but not vice versa.
Counterexamples to some variations on these statements can be found in the lists above. Specifically, Sierpiński space is normal but not regular, while the space of functions from R to itself is Tychonoff but not normal.
See also
- Collectionwise normal space – Property of topological spaces stronger than normality
- Monotonically normal space – Property of topological spaces stronger than normality
Citations
- ^ Willard, Exercise 15C
- ^ Engelking, Theorem 1.5.19. This is stated under the assumption of a T1 space, but the proof does not make use of that assumption.
- ^ "Why are these two definitions of a perfectly normal space equivalent?".
- ^ Engelking, Theorem 2.1.6, p. 68
- ^ Munkres 2000, p. 213
- ^ Willard 1970, pp. 100–101.
- ^ "separation axioms in nLab". ncatlab.org. Retrieved 2021-10-12.
- ^ Willard 1970, Section 17.
References
- ISBN 3-88538-006-4
- Kemoto, Nobuyuki (2004). "Higher Separation Axioms". In K.P. Hart; J. Nagata; J.E. Vaughan (eds.). Encyclopedia of General Topology. Amsterdam: ISBN 978-0-444-50355-8.
- ISBN 978-0-13-181629-9.
- Sorgenfrey, R.H. (1947). "On the topological product of paracompact spaces". Bull. Amer. Math. Soc. 53 (6): 631–632. .
- Stone, A. H. (1948). "Paracompactness and product spaces". Bull. Amer. Math. Soc. 54 (10): 977–982. .
- Willard, Stephen (1970). General Topology. Reading, MA: Addison-Wesley. ISBN 978-0-486-43479-7.