Download Internal Hom-Objects in the Category of Topological Spaces

Internal Hom-Objects in the Category of Topological
Spaces
Michael Hallam
Supervised by Professor Finnur Lárusson
The University of Adelaide
February 27, 2016
Abstract
It is an intriguing fact that internal hom-objects can fail to exist in the category of topological
spaces. Here we seek to understand this by providing examples which show the failure of the
compact-open and relatively compact-open topologies to be exponential.
Contents
1 Introduction
2 Background
2.1 Notation and Terminology
2.2 Statement of the Problem
2.3 Topological Framework . .
2.4 Categorical Framework . .
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3 Examples
3.1 The Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Square-Summable Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 The Co-Countable Reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1
Introduction
For ordinary sets A and B, denote by Hom(A, B) the set of functions from A to B. Given three
sets X, Y, T and a map g : T × X → Y we can construct a second map ḡ : T → Hom(X, Y ) called
the transpose of g, defined by ḡ(t)(x) = g(t, x). Every such map g induces a unique transpose in
this way. Conversely, every map ϕ : T → Hom(X, Y ) is the transpose of the map g : T × X → Y
defined by g(t, x) = ϕ(t)(x). So there is a bijection of hom-sets
Hom(T × X, Y ) ∼
= Hom(T, Hom(X, Y ))
given by g 7→ ḡ. This bijection is “natural” in the sense that how it is defined does not depend on
the sets X, Y and T . Category theory makes this precise with the notion of a natural bijection. In
this setting, X is called the source set, Y the target set and T the parameter set.
While this bijection may be trivial in the category of sets and plain maps, analogous results may fail
to hold in other familiar categories. In particular, such bijections can fail to exist in the category
of topological spaces and continuous functions. The simple definitions of topological spaces and
continuous functions give rise to a wealth of intricate theory, and have been tremendously successful.
This failure of bijections to exist is an unfortunate flaw and can be troublesome in algebraic topology,
for instance when the parameter space is the unit interval [0, 1].
It is the main purpose of this report to provide some examples which help to illustrate this failure.
In particular we show how for certain source and target spaces, the compact-open and relatively
compact-open topologies can fail to be exponential.
Section 2 introduces the problem and describes necessary and sufficient conditions for a solution
to exist. Section 3 is dedicated to the aforementioned examples, which are accompanied by related
results and generalisations. The required background theory is undergraduate topology. Some
category theory may help to understand the statement of the problem, however the examples can
be understood without any knowledge of category theory.
The proofs of Lemma 2.7 and Propositions 2.8, 3.1 and 3.2 were worked out by the researcher
(with help from Finnur on 3.2). To the best of our knowledge, all of the examples in Section 3 are
original.
We would like to thank the Australian Mathematical Sciences Institute for funding and making
this research possible.
2
Background
In this section we introduce the exponentiability problem for topological spaces, and study it from
the viewpoints of both topology and category theory. Necessary and sufficient conditions are given
for a topological space to be exponentiable.
2.1
Notation and Terminology
Given sets X and Y , by X ⊂ Y we mean that X is a subset of Y that is possibly equal to Y .
Topological spaces will often be referred to as spaces. The Cartesian product of any two spaces
will always be given the product topology. For a subset U of a space, we denote the interior of U
by U ◦ . For spaces X, Y we write C(X, Y ) for the set of continuous functions from X to Y – which
2
a priori is not a space – and call this a hom-set. A neighbourhood of a point x will be any set U
containing an open subset V which contains x. Given topologies T1 and T2 on a set, if T1 ⊂ T2 then
we say T1 is weaker than T2 and that T2 is stronger than T1 . The category of topological spaces
and continuous functions will be written Top. In a metric space, B(x, ) will denote the open ball
of radius centred at x.
2.2
Statement of the Problem
Given three spaces X, Y and T , we can turn a continuous function g : X × T → Y into a plain
function of sets ḡ : T → C(X, Y ) by defining ḡ(t)(x) = g(x, t). It does not make sense to ask
whether this map is continuous, since the set C(X, Y ) does not yet have a topology. Any set can
be given a topology – the trivial topology for example – however we would like to find a topology
on C(X, Y ) that is useful in some sense. This motivates the following definition:
Definition 2.1. A topology on C(X, Y ) is exponential if for every space T , the map g 7→ ḡ is a
bijection
C(T × X, Y ) ∼
= C(T, C(X, Y )).
A space X is exponentiable if for every space Y , the set C(X, Y ) has an exponential topology.
So a topology on C(X, Y ) is exponential if every continuous map g : T ×X → Y induces a continuous
transpose ḡ : T → C(X, Y ), and moreover every continuous function T → C(X, Y ) arises as the
transpose of a continuous function T × X → Y . We will see that exponential topologies are in fact
unique.
This definition of an exponential topology pins down the key properties of structured function sets
found in other familiar categories, such as the category of vector spaces. The key issue with which
we are concerned is:
It is not always possible to give the set C(X, Y ) an exponential topology.
In other words, not every topological space is exponentiable. This poses a rather nasty problem
for the category of topological spaces – a problem that many familiar categories do not share.
How and why does this failure occur? The problem has been well studied, and necessary and
sufficient conditions have been found for an exponential topology to exist. These are discussed in
the following section.
2.3
Topological Framework
The following terminology is is borrowed from [1] and is useful for discussing topologies on function
spaces. With the same notation as in Section 2.2, we have:
Definition 2.2. A topology on C(X, Y ) is weak if continuity of g implies continuity of ḡ, and strong
if continuity of ḡ implies continuity of g.
It follows that a topology on C(X, Y ) is exponential if and only if it is both weak and strong. The
trivial topology is an example of a weak topology, and the discrete topology is an example of a
strong topology. The following proposition, whose proof may be found in [1, Lemma 2.2], justifies
the above language.
Proposition 2.3.
• Any topology weaker than a weak topology is also weak.
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• Any topology stronger than a strong topology is also strong.
• Any weak topology is weaker than any strong topology.
This tells us that if an exponential topology on C(X, Y ) exists then it is unique, for if T1 and
T2 are both exponential topologies then by the third point we have T1 ⊂ T2 and T2 ⊂ T1 . The
exponential topology is then the strongest weak topology and the weakest strong topology. The
following proposition, whose proof is borrowed from [1, Lemma 2.1], gives a necessary and sufficient
criterion for a topology on C(X, Y ) to be strong.
Proposition 2.4. A topology on C(X, Y ) is strong if and only if the evaluation map φ : C(X, Y ) ×
X → Y defined by φ(f, x) = f (x) is continuous.
Proof. The transpose of the evaluation map is the identity map of C(X, Y ), which is continuous
regardless of the topology on C(X, Y ). So if the identity map induces a discontinuous evaluation
map then the topology on C(X, Y ) is not strong. Conversely, suppose that the evaluation map is
continuous. Given a continuous map ḡ : T → C(X, Y ), the corresponding map g : T × X → Y is
given by
g = φ ◦ (ḡ × 1X )
which is a composition of continuous functions and is therefore continuous. Hence the topology on
C(X, Y ) is strong.
This will be very useful when we look at some examples in the next section, mainly because a weak
topology on C(X, Y ) will fail to be exponential if and only if the evaluation map is discontinuous.
The following definition is crucial in the characterisation of exponentiable spaces.
Definition 2.5. A topological space X is core-compact if for every open set U and every point
x ∈ U , there exists an open set V ⊂ U containing x such that every open cover of U has a finite
subcover of V .
For arbitrary subsets U and V such that every open cover of U has a finite subcover of V we write
V U , and say that V is way below U .
For Hausdorff spaces, the condition of core-compactness reduces to a much more familiar property.
Definition 2.6. A topological space X is locally compact if every open neighbourhood U of a point
x contains a compact neighbourhood K of x.
In other words, X is locally compact if every point of X has a neighbourhood base of compact sets.
Examples of locally compact spaces include all discrete spaces, all trivial spaces, and all topological
manifolds. Local compactness can often be disproved by showing that a point has no compact
neighbourhood. We say that local compactness fails at a point x if there is a neighbourhood U of x
containing no compact neighbourhood of x. Be warned that some sources give different definitions
of local compactness, and these may not be equivalent for all spaces.
We wish to show that the conditions of local compactness and core-compactness are the same for
Hausdorff spaces, and to do this we will need a lemma:
Lemma 2.7. If X is a Hausdorff space and V U , then V̄ ⊂ U .
Proof. Suppose to the contrary that there is some point x ∈ V̄ \U . Since X is Hausdorff, for each
point y ∈ U there are disjoint open neighbourhoods Ay of x and By of y. We thus obtain an open
n
cover
Tn (By )y∈U of U which then has a finite subcover (Byi )i=1 of V , since V is way below U . But
i=1 Ayi is then a neighbourhood of x disjoint from V – contradiction.
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Proposition 2.8. A Hausdorff space is core-compact if and only if it is locally compact.
Proof. Let X be a locally compact Hausdorff space and U be an open neighbourhood of a point x.
Since X is locally compact, there is some compact neighbourhood V of x which is a subset of U .
Then V ◦ is an open subset of U containing x such that every open cover of U has a finite subcover
of V ◦ . Hence X is core-compact (this implication holds for non-Hausdorff spaces as well).
Conversely, suppose that X is Hausdorff and core-compact. Let U be any open neighbourhood of x.
Then by core-compactness there are open neighbourhoods V and W of x such that W V U .
We will show that W̄ is compact. Let U be an open cover of W̄ . Adding the open set U \W̄ to the
cover, we get an open cover U 0 of U . This then has a finite subcover of V , which is also a finite
subcover of W̄ by Lemma 2.7. If necessary, remove U \W̄ from the U 0 -subcover of W̄ to obtain a
finite U-subcover of W̄ .
For core-compact spaces, the relation has a nice property known as interpolation, meaning that
for an arbitrary subset W and an open set U with W U , there is an open set V such that
W V U – see [4, Lemma 1.2] for a proof. Knowing this fact, it is not hard to adapt the above
proof to show that if U is open in a Hausdorff space and W U then there is a compact set K
such that W ⊂ K ⊂ U , namely K = W̄ . So any set that is way below an open set in a Hausdorff
space must be precompact.
Given spaces X, Y and subsets A ⊂ X and W ⊂ Y , we denote by M (A, W ) the set of continuous
functions X → Y which map A into W . That is,
M (A, W ) = {f ∈ C(X, Y ) : f (A) ⊂ W }
= {f ∈ C(X, Y ) : A ⊂ f −1 (W )}.
Definition 2.9. The compact-open topology on C(X, Y ) is the unique topology generated by the
subbasis of sets M (A, W ), for each compact subset A of X and each open subset W of Y .
Since each singleton of X is compact, this topology is stronger than the topology of pointwise
convergence. In [3, Lemma 1], Fox proved the following proposition:
Proposition 2.10. The compact-open topology on C(X, Y ) is weak.
Therefore the compact-open topology is exponential if and only if it is strong, if and only if the
evaluation map is continuous. The following result suggests that for Hausdorff spaces, the compactopen topology is a likely candidate for an exponential topology — see [1, p. 11].
Proposition 2.11. If X is an exponentiable Hausdorff space then the exponential topology on
C(X, Y ) is the compact-open topology.
The examples in the next section will demonstrate the failure of the compact-open topology to be
exponential for certain Hausdorff source spaces. More generally, for non-Hausdorff spaces we have:
Definition 2.12. The relatively compact-open topology on C(X, Y ) is defined by taking a subbasis
of sets
R(A, W ) = {f ∈ C(X, Y ) : A f −1 (W )}
where A ranges over the subsets of X and W ranges over the open subsets of Y .
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In this definition the subsets A are arbitrary subsets of X. In [1, Theorem 5.3], the subsets A
are required to be open. While the topology given in Definition 2.12 is generally stronger, the
two topologies agree when the space X is core-compact, and this is an immediate consequence of
the interpolation property mentioned earlier. The next proposition relates the compact-open and
relatively compact-open topologies. See [4, Proposition 1.1] for a proof.
Proposition 2.13. The relatively compact-open topology is stronger than the compact-open topology. If X is locally compact then the compact-open topology and the relatively compact-open topology
on C(X, Y ) are the same.
Finally, we have from [1, Theorem 5.3] that:
Proposition 2.14. If X is an exponentiable space then the exponential topology on C(X, Y ) is the
relatively compact-open topology.
So now we have a full characterisation of exponentiable spaces and exponential topologies on function spaces with exponentiable sources. Among Hausdorff spaces, the exponentiable spaces are
precisely the locally compact spaces, and the exponential topologies on hom-sets with locally compact sources are the compact-open toplogies. Among arbitrary spaces, the exponentiable spaces are
precisely the core-compact spaces, and the exponential topologies on hom-sets with core-compact
sources are the relatively compact-open topologies.
Before we give examples that demonstrate the failure of the compact-open and relatively compactopen topologies to be exponential, there will be a brief interlude in which we describe the problem
of exponentiability in terms of category theory. For the reader who is not familiar with category
theory, this section may be safely skipped.
2.4
Categorical Framework
The exponentiability problem for topological spaces has strong ties to category theory. If a space
X is exponentiable then given any spaces T and Y , we have a bijection
C(T × X, Y ) ∼
= C(T, C(X, Y ))
given by g 7→ ḡ. This bijection is clearly natural in T and Y , so this means that the endofunctor
− × X on Top has a right adjoint given by C(X, −).
On the other hand, if the functor − × X has a right adjoint G : Top → Top then we have a
bijection
C(T × X, Y ) ∼
= C(T, GY )
that is natural in T and Y . Letting T = {1}, the one-point space, we see that
C({1} × X, Y ) ∼
= C({1}, GY ).
Noting the obvious natural bijections C(X, Y ) ∼
= C({1} × X, Y ) and C({1}, GY ) ∼
= U (GY ), where U
is the forgetful functor taking a topological space to its underlying set, we have C(X, Y ) ∼
= U (GY ).
If we endow C(X, Y ) with the topology induced by this bijection, then we have a bijection
C(T × X, Y ) ∼
= C(T, C(X, Y ))
that is natural in T and Y since we have only used natural bijections thus far.
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A right adjoint of − × X is often written (−)X . In the case where C(X, Y ) has an exponential
topology, we say that C(X, Y ) is an internal hom-object in the category of topological spaces – the
reason being that it is a hom-set which lands within the same category, Top.
The failure of some topological spaces to be exponentiable prevents Top from being a cartesian
closed category, that is a category with finite products and exponentials. Cartesian closed categories
have many nice properties and there is already much theory developed around them. Were the
category of topological spaces cartesian closed, we could use this extra machinery to solve problems
and develop theorems. Since Top is not cartesian closed, it is common practice to restrict attention
to full subcategories that are cartesian closed – see [2, p. 1] for examples. Such subcategories are
called convenient categories of topological spaces. A common choice is the category of compactly
generated Hausdorff spaces.
3
Examples
Here we give some examples which help to illustrate why some spaces fail to be exponentiable. The
examples are built upon with various propositions.
3.1
The Rational Numbers
Take (Q, | · |) as the source space. This is just Q with the subspace topology inherited from R
with the standard topology. Since it is a metric space, Q will fail to be exponentiable if and only
if it is not locally compact. Suppose that a point x ∈ Q has a compact neighbourhood U . This
neighbourhood is then the restriction to Q of a neighbourhood V of x in R, and there is some > 0
such that (x − , x + ) ⊂ V . Let ξ ∈ (x − , x + ) be irrational, and choose a sequence (rn ) of
rational numbers in V converging to ξ in R. Considered as a sequence in U ⊂ Q, this sequence has
no convergent subsequence, hence U is not compact and so Q is not locally compact.
Now that we know Q is not exponentiable, we want to choose a suitable target space to demonstrate
the failure of the compact-open topology to be exponential. Let R be the target space. We give
C(Q, R) the compact-open topology and seek to show that the evaluation map φ : C(Q, R) × Q → R
is discontinuous. Consider the open subset W = (0, 1) of R, and let (f, x) ∈ φ−1 (W ). We will show
that every neighbourhood of (f, x) in C(Q, R) × Q contains a point (g, y) with φ(g, y) 6∈ W .
Let G be a neighbourhood of (f, x). Without loss of generality, we may take
" n
#
\
G=
M (Ai , Wi ) × [ (x − , x + ) ∩ Q ]
i=1
T
where > 0, the Ai ⊂ Q are compact, and the Wi ⊂ R are open. Any function g ∈ ni=1 MS
(Ai , Wi )
necessarily maps Ai into Wi for each i. This restricts our ability to define g on the set A = ni=1 Ai .
Since each Ai is compact, A is also compact. Therefore, for any irrational ξ ∈ (x − , x + ), there
exists some neighbourhood U ⊂ R of ξ containing no elements of A. Otherwise we could obtain
a Cauchy sequence of numbers in A ⊂ Q converging to ξ in R which would have no convergent
subsequence in Q, contradicting the compactness of A.
Now, choose a rational number δ > 0 such that (ξ − δ, ξ + δ) ⊂ U . Let V = (ξ − δ, ξ + δ) ∩ Q and
define
(
f (z) z ∈ Q\V
g(z) =
2
z ∈ V.
7
T
Then g is continuous by the pasting lemma, and contained in ni=1 M (Ai , Wi ) since it agrees with
f on the set A. Since V ∩ (x − , x + ) is non-empty, there exists some point y ∈ (x − , x + ) ∩ Q
such that φ(g, y) = g(y) = 2 6∈ W . Noting that (g, y) ∈ G, this shows that the evaluation map is
discontinuous.
Note that the choice of the target space Y almost does not matter in this case. Provided there
exists a non-empty open subset W of Y that is not all of Y , the above argument will still hold.
This amounts to requiring that Y is a non-trivial space, so the compact-open topology on C(Q, Y )
fails to be exponential for most spaces Y .
We note here that this is not a general feature of spaces that are not exponentiable:
Proposition 3.1. If X is a connected space and D = {0, 1} is the two-point discrete space, then
the compact-open topology on C(X, D) is exponential.
Proof. We give C(X, D) the compact-open topology and show this topology is strong. Since X is
connected there are precisely two continuous functions X → D, namely f ≡ 0 and g ≡ 1. If x is
any point in X then
M ({x}, {0}) = {h ∈ C(X, D) : h(x) = 0} = {f },
M ({x}, {1}) = {h ∈ C(X, D) : h(x) = 1} = {g}.
These sets are open, so the compact open topology on C(X, D) is the discrete topology which is
strong. Hence the compact-open topology is exponential.
This holds regardless of whether X is exponentiable.
3.2
Square-Summable Sequences
For this example, we take `2 as the source space. This is the space of real square-summable
sequences
(
)
∞
X
2
`2 = (xn ) :
xn < ∞
n=1
with the metric
v
u∞
uX
d((xn ), (yn )) = t (xn − yn )2 .
n=1
Given any sequence (xn ) ∈ `2 and any > 0, the sequence of sequences
(x1 + , x2 , x3 , ...)
2
(x1 , x2 + , x3 , ...)
2
(x1 , x2 , x3 + , ...)
2
..
.
is in B((xn ), ) and has no convergent subsequence. It follows that (xn ) has no compact neighbourhood, hence `2 is not locally compact.
8
Again we take R as the target space, endow C(`2 , R) with the compact-open topology and take this
as our parameter space. We will show that the identity on C(`2 , R) does not induce a continuous
evaluation map. Let W = (0, 1) ⊂ R and take f ∈ C(`2 , R) with f (0) ∈ W , where 0 = (0, 0, 0, ...).
Let G be a neighbourhood of (f, 0). Assume without loss of generality that
" n
#
\
G=
M (Ai , Wi ) × B(0, )
i=1
S
where > 0, the Ai ⊂ `2 are compact and the Wi ⊂ R are open. Then A = ni=1 Ai is compact.
Hence for every δ > 0 there exists some N ∈ N such that for each x ∈ A, |xn | < δ if n ≥ N .
Otherwise we may obtain a sequence of elements in A with no convergent subsequence, contradicting
the compactness of A. Take δ = /2 and let y = (0, ..., 0, 3/4, 0, ...) ∈ B(0, ), where 3/4 is the
(N + 1)-th term of the sequence. Then d(y, x) ≥ /4 for each x ∈ A. Now define
(
f (x)
d(x, y) ≥ /4
g(x) =
8
4
( 4 − d(x, y)) + d(x, y)f (x) d(x, y) ≤ /4.
Tn
Then g is continuous, g ∈ i=1 M (Ai , Wi ) and g(y) = 2 6∈ W . So φ is not continuous.
This example closely follows the previous example, although note that we cannot freely change the
target space as before. This is due to our use of addition and multiplication in R in the definition
of g. If we changed the target space then we would have to find a different g, and as Proposition
3.1 shows, this may not even be possible since `2 is connected. This example lends itself to the
following generalisation:
Proposition 3.2. If X is a completely regular Hausdorff space containing a point x with no compact
neighbourhood, then the compact-open topology on C(X, R) is not exponential.
Proof. Give C(X, R) the compact-open topology. Consider the continuous function f ≡ 1 on X.
Then (f, x) ∈ φ−1 (W ) where W = (3/2, 1/2). Take a neighbourhood
" n
#
\
G=
M (Ai , Wi ) × U
i=1
of (f, x), where the Ai ⊂ S
X are compact and the Wi ⊂ R are open. Since x has no compact
neighbourhood and A = ni=1 Ai is compact, there must be some point y ∈ U \A. As X is
completely regular there is a continuous function g : X → R such that g(A) = {1} and g(y) = 0.
Since g agrees with f on the set A, we have (g, y) ∈ G but φ(g, y) 6∈ W . Hence the evaluation map
is not continuous, so the topology on C(X, Y ) is not exponential.
3.3
The Co-Countable Reals
Take R with the co-countable topology (written Rcc ) as the source space. Let S be the space whose
underlying set is {0, 1} such that {1} is open and {0} is not. This space is known as the Sierpinski
space. It turns out that a space X is exponentiable if and only if C(X, S) has an exponential
topology – see [1, pp. 5–6]. So we will take S as our target space.
Since Rcc is non-Hausdorff, it will fail to be exponentiable if and only if it is not core-compact.
Consider the open neighbourhood Rcc of 0. Let W be another open neighbourhood of 0. Since W
is non-empty and open, it is the complement of a countable set. So we can find a sequence (an ) of
9
S
distinct points in W . Let A = {an }. The family of sets ({an } ∪ (Rcc \A)) is then an open cover
of Rcc with no finite subcover of W . Hence Rcc is not core-compact.
We give C(Rcc , S) the relatively compact-open topology. To check continuity of the evaluation map
φ, we need only check that φ−1 (1) is open. Let f ≡ 1 on Rcc and take a neighbourhood
" n
#
\
G=
R(Ai , Wi ) × U
i=1
of (f, 0), where Ai ⊂ Rcc , the Wi ⊂ S are open and U is an open neighbourhood of 0. If h ∈
R(Ai , Wi ) then Ai h−1 (Wi ). The only way this is possible
Sn is if Ai is finite (by a similar argument
in the proof that Rcc is not core-compact). Hence A = i=1 Ai is finite. Since U is non-empty and
open, it follows that there is some y ∈ U \A. Define
(
1 x 6= y
g(x) =
0 x = y.
Then (g, y) ∈ G but φ(g, y) 6= 1 and so the evaluation map is discontinuous.
References
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Topology Proc. 26 (2001/02), no. 2, 545–564.
[2] Escardó, Martı́n, Jimmie Lawson, and Alex Simpson. Comparing Cartesian closed categories of
(core) compactly generated spaces. Topology Appl. 143 (2004), no. 1-3, 105–145.
[3] Fox, Ralph H. On topologies for function spaces. Bull. Amer. Math. Soc. 51 (1945) 429–432.
[4] Lowen-Colebunders, Eva and Günther Richter. An elementary approach to exponential spaces.
Appl. Categ. Structures 9 (2001), no. 3, 303–310.
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