Perfect set

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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 Gδ 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 is a connected perfect space, while the Cantor space 2ω and Baire space ωω 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 be a family of topological spaces. As for any local property, the disjoint union is perfect if and only if every is perfect.

The Cartesian product of a family is perfect in the product topology if and only if at least one of the following holds:

(i) At least one is perfect.

(ii) .

(iii) The set of indices such that 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 -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 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 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:

See also


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