Perfect space

In mathematics, in the field of topology, perfect spaces are spaces that have no isolated points. In such spaces, every point can be approximated arbitrarily well by other points - given any point and any topological neighborhood of the point, there is another point within the neighborhood.

The term "perfect space" is also used, incompatibly, to refer to other properties of a topological space, such as being a G_δ space. Context is required to determine which meaning is intended.

In this article, a space which is not perfect will be referred to as imperfect.

Examples and nonexamples

The real line $mathbb\{R\}$ is a connected perfect space, while the Cantor space 2^ω and Baire space ω^ω are perfect, totally disconnected spaces of topological dimension 0.

Any nonempty set admits an imperfect topology: the discrete topology. Any set with more than one point admits a perfect topology: the indiscrete topology.

Imperfection of a space

Define the imperfection of a topological space to be the number of isolated points. This is a cardinal invariant -- i.e., a mapping which assigns to each topological space a cardinal number such that homeomorphic spaces get assigned the same number.

A space is perfect if and only if it has imperfection zero.

Closure properties

Every nonempty perfect space has subsets which are imperfect in the subspace topology, namely the singleton sets. However, any open subspace of a perfect space is perfect.

Perfection is a local property of a topological space: a space is perfect if and only if every point in the space admits a basis of neighborhoods each of which is perfect in the subspace topology.

Let $\{X\_i\}\_\{i\; in\; I\}$ be a family of topological spaces. As for any local property, the disjoint union $coprod\_i\; X\_i$ is perfect if and only if each $X\_i$ is perfect.

The Cartesian product of a family $\{X\_i\}\_\{i\; in\; I\}$ is perfect in the product topology if and only if at least one of the following holds:

(i) At least one $X\_i$ is perfect.

(ii) $I\; =\; emptyset$.

(iii) The set of indices $i\; in\; I$ such that $X\_i$ has at least two points is infinite.

A continuous image, and even a quotient, of a perfect space need not be perfect. For example, let "X" = R-{0}, let "Y" = {1, 2} given the discrete topology and let "f" be a function defined such that "f(x)" = 2 if x > 0 and "f(x)" = 1 if x < 0. However, every image of a perfect space under an injective continuous map is perfect.

Connection with other topological properties

It is natural to compare the concept of a perfect space -- in which no singleton set is open -- to that of a T₁ space -- in which every singleton set is closed.

A T₁ space is perfect if and only if every point of the space is an $omega$-accumulation point. In particular a nonempty perfect T₁ space is infinite.

Any connected T₁ space with more than one point is perfect. (More interesting therefore are disconnected perfect spaces, especially totally disconnected perfect spaces like Cantor space and Baire space.)

On the other hand, the set endowed with the topology $\{\; emptyset,\; \{\; circ\; \},\; X\; \}$ is connected, T₀ (and even sober) but not perfect.

Suppose "X" is a homogeneous topological space, i.e., the group $operatorname\{Aut\}(X)$ of self-homeomorphisms asks transitively on "X". Then "X" is either perfect or discrete. This holds in particular for all topological groups.

A space which is of the first category is necessarily perfect.

Perfect spaces in descriptive set theory

Classical results in descriptive set theory establish limits on the cardinality of non-empty, perfect spaces with additional completeness properties. These results show that:
* If "X" is a complete metric space with no isolated points, then the Cantor space 2^&omega; can be continuously embedded into "X". Thus "X" has cardinality at least $2^\{aleph\_0\}$. If "X" is a separable, complete metric space with no isolated points, the cardinality of "X" is exactly $2^\{aleph\_0\}$.
* If "X" is a locally compact Hausdorff space with no isolated points, there is an injective function (not necessarily continuous) from Cantor space to "X", and so "X" has cardinality at least $2^\{aleph\_0\}$.

ee also

*Finite intersection property
*Derived set (mathematics)
*Subspace topology

References

* Citation
last1=Kechris
first1=A. S.
author1-link=Alexander S. Kechris
title=Clasical Descriptive Set Theory
publisher=Springer-Verlag
location=Berlin, New York
isbn=0387943749 ISBN 3540943749
year=1995
* Citation
last1=Levy
first1=A.
author1-link=Azriel Levy
title=Basic Set Theory
publisher=Springer-Verlag
location=Berlin, New York
year=1979
* Citation
editor1-last=Pearl
editor1-first=Elliott
title=Open problems in topology. II
publisher=Elsevier
isbn=978-0-444-52208-5; 978-0-444-52208-5
id=MathSciNet | id = 2367385
year=2007