topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
What is called Cantor space, after Georg Cantor, is the topological space obtained from the closed interval by
removing the middle third, retaining only and ;
removing from these two pieces their middle thirds, to retain the pieces , , , ;
and so ever on.
Cantor space is a fundamental object of descriptive set theory; some indications of its use may be found at Polish space. Among its applications is a simple construction of a โspace-filling curveโ (q.v.).
Traditionally the Cantor space was conceived of
but of course it may also be described
in itself.
In brief, Cantor space may be abstractly described as the topological product of countable many copies of the discrete space . In more concrete detail:
Recall that a binary digit is either or ; the set (or discrete space) of binary digits is the Boolean domain .
A point in Cantor space is an infinite sequence of binary digits. Accordingly, Cantor space may be denoted , since its set of points is a function set.
An open in Cantor space is a collection of finite sequences of binary digits (that is a subset of the free monoid ) such that:
If and is an extension of (that is with possibly additional digits added to the end), then ;
If and (where is the immediate extension of by the digit ), then .
A point belongs to an open if, for some in , is an extension of .
An alternative characterization of Cantor space is as the terminal coalgebra for the endofunctor on Top, .
Traditionally, Cantor space is understood as a topological space. We start with the points, as defined above, then specify which sets of points are open. Although there are other ways to state which sets are open, we may define a set to be open if it is the set of points that belong to some open as defined above.
A newer approach is to understand Cantor space as a locale. Then we start with the opens and define an order relation on them to define a frame. In this case, the order relation is the obvious one, that if as subsets of . Then the points come for free, and correspond precisely to the points as defined above.
In classical mathematics, these two approaches are equivalent; a point is determined by its opens, and an open is determined by its points. The theorem that a point is determined by its opens (so that Cantor space, as a topological space, is sober) is valid internal to any pretopos with an exponentiable natural numbers object; as such, it applies even in predicative and constructive mathematics. However, the theorem that an open is determined by its points (so that Cantor space, as a locale, is topological) is equivalent to the fan theorem; it is true in some pretoposes and accepted by some schools of constructivism but false in other pretoposes and rejected, or even refuted, by other constructivists.
When the fan theorem is not valid, the localic approach is probably better; it allows more of the useful properties of Cantor space to hold.
Cantor space is usually conceived of as a topological subspace of the real line:
Write for the the discrete topological space with two points. Write for the product topological space of a countable set of copies of this discrete space with itself (i.e. the corresponding Cartesian product of sets equipped with the Tychonoff topology induced from the discrete topology of ).
Then consider the function
which sends an element in the product space, hence a sequence of binary digits, to the value of the power series as shown on the right.
One checks that this is a continuous function (from the product topology to the Euclidean metric topology on the closed interval). Moreover with its image equipped with its subspace topology, then this is a homeomorphism onto its image:
This image is the Cantor space as a subspace of the closed interval.
From the localic perspective, a continuous map is given by a homomorphism of frames in the opposite direction. Given an open in (as a binary relation on rational numbers, as described at locale of real numbers), this is mapped to the open in Cantor space such that if and only if
One then checks that this is an embedding.
I should check this some day; for the moment, I am taking it on faith. โToby
In either case, the idea is:
A point of Cantor space corresponds to a number written in base with infinitely many digits, using only the digits and (which are the options for when ); while
An open corresponds to a union of intervals, each of which is given by approximating a number in base to a finite number of digits, using only the digits and .
One sometimes speaks of the Cantor set to stress that one is considering Cantor space as a subspace of the real line.
As we can also consider Cantor space as a product space of countably many copies of , which carries a group structure, we can view Cantor space as a topological group. In particular, it is a homogeneous space (its group of self-homeomorphisms acts transitively on the space).
Cantor space, especially in its guise as a subspace of the real line, is quite famous; see Wikipedia. Here are some headline properties:
Cantor space is a compact Hausdorff space. (For the topological space, this statement is again equivalent to the fan theorem; for the locale, it holds regardless.)
Cantor space is totally disconnected.
Thus Cantor space is a Stone space.
Cantor space is metrizable, and every compact metrizable space is a quotient space of Cantor space (see Theorem below).
As a subspace of , the Cantor set is perfect and uncountable but of Lebesgue measure zero.
The Cantor set is a precisely self-similar fractal with Hausdorff dimension .
A topological space is homeomorphic to Cantor space if and only if it is nonempty, compact, totally disconnected, metrizable, and perfect.
This result is sometimes called Brouwerโs theorem. It can be seen from the perspective of Stone duality, where the dual result is that any two countable atomless Boolean algebras are isomorphic; this dual result can be proven by a back-and-forth argument.
The one-point compactification of a space that is second-countable locally compact Hausdorff, totally disconnected and perfect, is homeomorphic to Cantor space (provided is not itself compact).
is also second-countable, compact Hausdorff and therefore compact regular, and so by the Urysohn metrization theorem it is compact metrizable. The point at infinity is not isolated since we assume is not compact, so is perfect. If is any open neighborhood of , so for some compact , then we claim there exists a clopen that contains ; in that case contains a clopen whence is the quasi-component of (hence also the connected component since weโre in a compact Hausdorff space). But the argument here shows that for each there is a clopen neighborhood of contained in ; finitely many of these clopens cover , and the claim follows by considering their union.
It follows from this result that all such spaces are homeomorphic: they all have Cantor space as their one-point compactifications, and so they are all homeomorphic to the space obtained obtained by removing a single point from Cantor space. This applies for example to spaces obtained by removing a finite number of points from Cantor space.
Cantor space is also a โuniversalโ compact metric space in the following sense.
(Hausdorff-Alexandroff) Every compact metric space is a continuous image of Cantor space.
This implies that every compact metric space is a quotient space of Cantor space, since a surjective map between compact Hausdorff spaces is a closed surjection, and closed surjections are quotient maps.
First, every compact metric space is separable: has a countable dense set . Assume, as we may, that the metric is valued in . Then the map to the Hilbert cube, defined by
(a type of restricted โYoneda embeddingโ, regarding metric spaces as enriched categories), is continuous and maps onto a closed subspace of . As mentioned at Peano curve, there is a continuous surjection . Taking the pullback in Top
we see that to produce a continuous surjection , it suffices to exhibit a continuous surjection .
In fact, every closed subspace admits a retraction. There is a clever trick for seeing this: represent Cantor space instead as the subspace of whose points, when written in base , have just 's and 's in their representation. This subspace has the geometric property that if , then . As a result, for we have only if and so: given a closed subspace of , there is for each a unique element such that . The assignment is continuous (in fact a locally constant function on , and continuous on as is easily seen) and provides the desired retraction.
Named after Georg Cantor.
Friedhelm Waldhausen, p. 3-4 in Topologie (pdf)
Proof Wiki, Cantor Space as Countably Infinite Product
Last revised on June 24, 2024 at 11:01:25. See the history of this page for a list of all contributions to it.