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
This page gives an elementary description of the locale of real numbers, that is the localic real line. The development is manifestly constructive and even predicative over the natural numbers (although we are somewhat careless with the language and do not always point out when a set may predicatively be a proper class). Ideally, we will show that our construction satisfies the seven ‘headline properties’ of the real line described by Bauer & Taylor (although so far we cover only the Heine–Borel theorem).
The exposition here is pretty much Toby Bartels's own work, although of course the basic ideas are well known to many. In particular, the Zigzag Lemma is Bartels's as far as they know (but it's not very deep, just bookkeeping). The version of Cousin's Theorem? that appears here may also be original.
To describe the locale of the real numbers, we need first of all to describe what an open (in the axiomatic sense, equivalent to an open sublocale) in the real line is. The key insight is that an open subspace is determined by the open intervals of rational numbers that it contains. This is analogous to the key insight of Richard Dedekind's definition of real number: that a point in the real line is determined by which open intervals of rational numbers that it belongs to. (Once one is talking about rational numbers, things are manageable.)
An open in the real line is a binary relation on the rational numbers that satisfies the four properties listed below. Intuitively, we have iff the open interval is contained in the corresponding open set.
If , then .
If , then .
If , then .
If whenever and , then .
In practice, rather than using relation-like symbols for opens, we mimic the set-theoretic notation and terminology from point-set topology. So if is an open in the real line, then we write to mean that is related to through and say that contains the interval from to and that this interval is contained in .
In this notation, and writing for and , and for and , these requirements say:
If , then .
If and , then .
If and , then .
If for all , then .
Property (1) is motivated because is empty whenever . Property (2) is motivated because inclusion is transitive. Property (3) is motivated because if , then . Property (4) is motivated because .
The really interesting property is property (3). It immediately generalizes as follows:
then .
We call the combined hypothesis of this property a zigzag; each hypothesis is a zig, and each hypothesis is a zag. To indicate the length of a zigzag, we will count the zigs; the zigzag above has zigs (and zags). A typical nondegenerate zigzag with zigs is shown below; it consists of overlapping open intervals, each of which belongs to a given open set; we are motivated to conclude that the entire interval from to belongs to that open set.
Thus the zigzag property for is property (3), the zigzag property for is trivial (if , then ), and the zigzag property for may be proved by induction.
We can incorporate property (2) into this by saying:
then .
Then you can think of property (1) as the zigzag property for . Alternatively, we can incorporate property (4) by saying:
then .
Or by making both modifications, we can stuff the entire definition into a single statement.
A zigzag may not look like the picture above; instead of an orderly progression, the zigs and zags may be wild swings that are undone by other zags and zigs. It is sometimes convenient to replace a zigzag by a more orderly one. Of course, this is easy if all of the zigs belong to a single open, since (by property (3) of the definition of an open), the whole zigzag can be replaced by a single zig. But we'll also want to consider zigzags where the zigs are taken from different opens. The following definitions make precise what kind of zigzag we'll want, and then there will be a lemma that we can indeed have this.
Given rational numbers and and a positive natural number , a -tuple of rational numbers forms a zigzag of length from to if we have:
A zigzag of positive length is orderly if we additionally have:
As a technicality, if , then we also count the empty -tuple as an orderly zigzag of length from to . (Notice that no zigzag of positive length can be orderly from to when .)
Given additionally a collection of opens, a zigzag (orderly or not) has zigs from if every pair is contained in at least one member of .
Now we can replace any zigzag with an orderly zigzag with zigs from the same opens:
(Zigzag Lemma)
Given rational numbers and , a natural number , a collection of opens, and a zigzag from to of length with zigs from , there exists an orderly zigzag from to of length at most with zigs from .
If , then we use a zigzag of length . Otherwise, we assume that and prove the lemma by induction on . If , then the original zigzag is orderly since .
Now assume (as an inductive hypothesis) that the lemma holds for zigzags of length for some . A zigzag of length consists of a zig from some open in , a zag , and a zigzag of length from to with zigs from . Using the inductive hypothesis, replace the zigzag of length with an orderly zigzag of length from to with zigs from . We now have a zigzag of length from to with zigs from .
If and , then also , so this zigzag of length from to is orderly.
Next suppose that but . In this case, consider the largest value of such that ; if , then we can use the orderly zigzag of length from to , which is contained in the same open as was, since . If instead, then take the orderly zigzag from to , and precede it with the zig , to get a zigzag of length from to . This zigzag is orderly, because , and its zigs are from since is contained in the same open as was (since ). So either way, we have an orderly zigzag of length at most from to with zigs from .
Finally, suppose that . In this case, consider the largest value of such that , take the orderly zigzag from to , and change to to get a zigzag from to . If , then this is orderly since ; if , then this is orderly since . Either way, belongs to the same open as did (since ), so this orderly zigzag of length from to still has zigs from .
In any case, we have a zigzag of length at most from to with zigs from , and the induction is complete.
This lemma is used in the proofs of the infinite distributivity law and some results related to measure.
The opens form a sub-poset of the power set . This poset is in fact a frame, as we will now show. (It is not a subframe of the power set, since the joins are different. It is a sub-inflattice of the power set, although this seems to be a red herring at least for infinitary meets, since those are not part of the frame structure that we need.)
The top open, denoted , is the binary relation which is always true. Given two opens and , their meet in the poset of opens, denoted , is simply their conjunction, that is their intersection as subsets of . (In fact, given any collection of opens, their meet is their conjunction.) It is straightforward to check that and are opens and to prove that these are the desired meets. Intuitively, this all works because an open interval will be contained in the intersection of a family of open sets if and only if it is contained in each individual open set.
The bottom open, denoted , is the binary relation . That is, iff . It is easy to check that this is an open; it precedes every open by property (1). Intuitively, this corresponds to the empty subset of the real line; is empty if and only if . However, note that is not the empty subset of ; the notation follows our topological intuition rather than the algebra of relations.
Given two opens and , their join in the poset of opens, denoted , is defined as follows: if, whenever and , there exists a zigzag from to with zigs from and . It is immediate that this is an open in which and are both contained. Conversely, any open in which and are contained must contain this open , by properties (3) and (4).
More generally, given any collection (or family ) of opens, their join in the poset of opens, denoted (or ), is defined as follows: if and only if, whenever and , there exists a zigzag from to with zigs from . (Notice that counts even when is empty, using zigzags of length .) The same argument applies as before. Note that each individual zigzag has finitely many zigs, and therefore involves finitely many of the opens , even when taking the join of an infinite collection.
Finally, we must check the distributive law . That is, if directly through and through a zigzag of s for , then we need that through a zigzag in which each zig is related both through and through some . To prove this, start with the zigzag of s, and apply the Zigzag Lemma to get an orderly zigzag of s, so that each zig is bounded by and . Then these zigs hold for as well, by property (2). Therefore, we may interpret each zig using for some , proving the desired result.
This frame of opens, interpreted as a locale, is the locale of real numbers. As usual, we denote this locale with the same symbol as the top element of its frame, in this case . (Of course, the true etymology of the symbols runs in the other order.)
Given rational numbers and , the open interval may itself be interpreted as an open in the real line, also denoted , as follows: let hold if every rational number strictly between and (in that order) is also strictly between and (in that order). In other words, we interpret ‘’ literally as comparing subsets of . It is straightforward to check that this condition does indeed define an open. There is now a third way to interpret ‘’; interpreting both intervals as opens in the real line, this states that the first is contained in the second. But again, it is easy to check that this is equivalent; (in the set-theoretic sense) if and only if whenever . Notice that whenever .
We can actually generalize this somewhat. Given any set of rational numbers and any set of rational numbers, we may define the open as follows: let hold if every rational number strictly between and (in that order) is greater than some element of and less than some element of . If the infimum of and the supremum of exist in the usual sense as rational numbers, then this agrees with the previous paragraph. If instead or is the set of all rational numbers, then we write for or for . In general, we may interpret as an extended upper real and as an extended lower real. Classically, every extended upper or lower real is either a real number, , or ; only the converse holds constructively. Notice that .
Since we will refer to them below, we will state for the record the complete definitions of and for a rational number . We have iff every rational number strictly between and is less than , that is iff or . Similarly, we have iff every rational number strictly between and is greater than , that is iff or . A fortiori, if , and if .
We think of each open as defining an open subset of the real line, but we can equally well think of it as defining a closed subset. The difference between these perspectives is reflected in complementary criteria for when a real number belongs to the set. So to make sense of this, we must identify the points of the real line.
Recall that a real number may be defined as a pair of inhabited subsets of satisfying the following properties:
We define a point of the real line to be a real number in this sense. Given such a point and a rational number , we write to mean that and to mean that . If we wish to refer to and directly, we may call the lower set of and the upper set.
Given a point and an open , we say that belongs to , written , if for some and ; that is, contains an interval from some element of the lower set to some element of the upper set of . We have since its lower and upper sets are inhabited. If and , with and , then , so ; note that this argument fails for infinitary intersections. (The converse, that and if , is immediate.) Dually, suppose that , as shown by some zigzag (since is impossible when and , by 3). Applying condition (4) of the definition of real number to each zag, we have or , for each . Checking all possibilities, and knowing that and in any case, we must have and for some . Then we have for some , whichever corresponds to the th zig. (The converse, that if for some , is immediate.)
Therefore, each point defines a completely prime filter on the frame of all opens, which is the definition of a point in general locale theory. Conversely, given such a completely prime filter , let be the set of all rational numbers such that the open interval (when interpreted as an open in the real line as defined above) is in , and symmetrically let be the set of all such that .
Given a point and an open , we say that co-belongs to , written , if we never have , , and , which is precisely the negation of the property that . We think of this condition as defining a closed set to which does belong. Notice that if and only if for each , giving the desired behaviour for an arbitrary intersection of closed sets (which corresponds to union of open sets under de Morgan duality). We also have that always fails, and if or . To prove that or whenever , however, we must use excluded middle; constructively, closed sets don't behave well under union.
A related question is whether we can reconstruct from the set of points which belong to it. This should be equivalent to the fan theorem, which is classically true and also accepted by Brouwer's school of intuitionism, but refuted in the Russian school in which all real numbers are assumed to be computable. (I should check this.) Arguably, the real lesson of these logical technicalities is that we should remember that opens, not points, are the fundamental concept in a locale.
The classical Heine–Borel theorem, as a statement about sets of real numbers, may fail constructively; this is related to the comments above about the fan theorem. But the beauty of the localic approach is that Heine–Borel necessarily holds when interpreted as a statement about opens in the locale of real numbers. To state the theorem, we must define what it means for a collection of opens to cover the unit interval. We will give an ad-hoc definition, but this may also be derived from the general theory of closed sublocales which allows us to interpret the unit interval as a compact locale in its own right.
An open cover of the unit interval is a collection of opens in the real line such that is the join of , , and the members of .
(Heine–Borel)
Every open cover of the unit interval has a finite subcover.
The proof is almost embarrassingly simple. The key point is that the construction of joins in terms of zigzags involves only finite zigzags, even for an infinitary join.
Let be the join of , , and the members of . Since this equals , then in particular , and since , we get a corresponding zigzag involving finitely many zigs using finitely many of the members of . Let be the collection of these members of , and let be the join of , , and . Now if is any pair of rational numbers, we construct a zigzag showing directly that as follows: the zig , the zag , the zigzag from to , the zag , and the zig . This is always a valid zigzag, so . Therefore, the finite collection covers the unit interval.
This proof generalizes to any closed interval , for any upper real and any lower real. But note that we do not say ‘extended’ here; we need to find some rational number (analogous to in the proof above) smaller than and some rational number (analogous to above) larger than . So the Heine–Borel theorem applies only to bounded closed intervals.
Another generalization is Cousin's Theorem?:
(Cousin's)
Given any open cover of the unit interval, there is a partition such that each subinterval is covered by a single member of (in that ).
As in the previous proof, construct a zigzag from to with zigs from . Using the Zigzag Lemma , replace this with an orderly zigzag. Let be , let be the smallest left endpoint of a zig from satisfying , let be smallest left endpoint satisfying , and so on until none are left, and then finish with .
A corollary of this theorem, when the open cover is given by a function (so that ), is used (classically) to prove the uniqueness (indeed, non-vacuity) of the Henstock–Kurzweil integral, which could be used to define Lebesgue measure (see below).
The set of computable real numbers has measure zero in the sense that, given any positive number , there's a collection of open intervals (even with rational endpoints), such that every computable real number belongs to at least one member of , and the sum of the lengths of any finite list of distinct members of is less than . (We can do this by enumerating Turing machines.) This is a problem for a theory of Lebesgue measure in Russian constructivism, where every real number is computable.
From a localic perspective, however, while might cover all the points of the real line, it cannot cover all of the pointless parts (which need not be empty). Indeed, we have this result ruling out such shenanigans:
Given any open interval with rational endpoints and any collection of such intervals, if , then for any length , there must be a finite? (in the strictest sense) subcollection of such that
In other words, any cover of by intervals must have a total length (defined as the supremum of the lengths of finite subcollections) at least the length of .
Let with (say with and ). Then there is a zigzag from to with zigs from ; by the Zigzag Lemma , we may take this zigzag to be orderly, say of length . Each zig is from one of the intervals in , say . If these intervals are all distinct (which we can decide, since the endpoints are rational), then we're done, since
or if . If they're not distinct, then we instead use the smallest and largest that appears in a given interval, and the argument still works (possibly even with some overlap now).
This result can be strengthened to intervals without necessarily rational endpoints, by using the rational intervals that they contain, but this is not the most general statement either, and I think that what we really need to do is to develop a theory of measures of open subspaces (or perhaps even more general subspaces) and state a result about that.
The locale of real numbers is the classifying locale of the geometric theory of Dedekind real numbers.
There are two general ways to define functions on the locale of real numbers, via the rational numbers and via the Dedekind real numbers.
The first approach uses existing functions already defined on the rational numbers . Suppose that one has a locally uniformly continuous function . Then one could extend to a continuous map on the locale of real numbers.
The other approach is via using existing functions defined on the Dedekind real numbers . In particular, given any well-defined continuous function on the Dedekind real numbers, and a geometric morphism from Set to the sheaf topos , if for all Dedekind real numbers , , then the function lifts to a function on the locale of real numbers.
This allows us to first define real-valued functions normally on the set of Dedekind real numbers, and then lift them up to a real-valued function on the locale of real numbers, thus allowing us to define functions such as the exponential function, the sine, and the cosine which can't be directly defined using functions on the rational numbers.
Significantly, since the field operations on the discrete locale of rational numbers are all locally uniformly continuous on the , and the field operations are well-defined and continuous on the Dedekind real numbers , the field operations in both cases lift up to a field structure on the locale of real numbers, allowing one to do basic arithmetic on .
Simon Henry, Localic Metric spaces and the localic Gelfand duality, (arXiv:1411.0898)
Guillaume Raynaud, Fibred Contextual Quantum Physics (etheses:1685, pdf)
The locale of real numbers as the classifying locale of the geometric theory of two-sided Dedekind cuts:
An (impredicative) construction of the locale of real numbers can be found in section 5.3 of:
On lifting functions from the Dedekind real numbers to the locale of real numbers, see:
Madeleine Birchfield, Simon Henry, The field structure on the locale of real numbers (2022) [MO:q/434706]
Valery Isaev, Simon Henry, Localic maps given by series (MathOverflow)
Last revised on February 6, 2024 at 15:49:59. See the history of this page for a list of all contributions to it.