Perfect set

In mathematics, in the field of topology, a perfect set is a closed set with no isolated points, and a perfect space is any topological space with 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 zero-dimensional spaces.

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 every $X_i$ 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 $X = \{ \circ, \bullet \}$ endowed with the topology $\{ \emptyset, \{ \circ \}, X \}$ is connected, T₀ (and even sober) but not perfect (this space is called Sierpinski space).

Suppose X is a homogeneous topological space, i.e., the group $\operatorname{Aut}(X)$ of self-homeomorphisms acts 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 (so, similar to compactifiying a space, we can 'make' a space to be of the second category by taking the disjoint union with a one-point space).

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^ω 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 $2^{\aleph_0}$ .

References

Kechris, A. S. (1995), Classical Descriptive Set Theory, Berlin, New York: Springer-Verlag, ISBN 3540943749
Levy, A. (1979), Basic Set Theory, Berlin, New York: Springer-Verlag
edited by Elliott Pearl. (2007), Pearl, Elliott, ed., Open problems in topology. II, Elsevier, ISBN 978-0-444-52208-5, MR 2367385