A note on the category of c-spaces

We prove that the category of c-spaces with continuous maps is not cartesian closed. As a corollary the category of locally finitary compact spaces with continuous maps is also not cartesian closed.


Introduction
Many people have been trying to extend domain theory to general topological spaces, see [6,4,7,9,2].Directed spaces are introduced by Kou and Yu independently [12] in 2014 for generalizing the concept of Scott spaces, which is equivalent to that of T 0 monotone determined spaces introduced by Erné [5].In the same paper Kou and Yu proved that the category of directed spaces with continuous maps (DTop for short) is cartesian closed.There are many important directed spaces in domain theory, for instance locally finitary compact spaces are directed spaces; in particular c-spaces and Alexandroff spaces are directed spaces.Since the category of continuous domains is not cartesian closed, and since the position of the category of c-spaces in the category of directed spaces is similar to that of continuous domains in dcpos [3], a natural question arises: Is the category of c-spaces cartesian closed?In this short note, we answer this question in the negative.

Preliminaries
We refer to [8,1,9] for the standard definitions and notations in order theory, topology and domain theory.A partially ordered set D is called a dcpo if every directed subset of D has a supremum in D. A upper set U is called a Scott open set if for any directed set A ⊆ D, A ∈ U implies A intersects U .

7-2
A note on the category of c-spaces For a topological space X, we use O(X) to denote the lattice of open subsets of X.We require that all topological spaces are T 0 in this note.Let X be a T 0 space, the specializing order ≤ is defined as follows : x ≤ y if x belongs to the closure of point y.A topological space is a c-space if for any x ∈ X and any open neighbourhood U of x, there is a point y ∈ U such that x ∈ int (↑y).A space X is locally finitary compact if for any x ∈ X and its open neighborhood U , there is a finite subset F of U such that x ∈ int (↑F ).
Let X be a T 0 space and ≤ the specialization order over X.A topological space X is called a Scott space if (X, ≤) is a dcpo and the topology on X is equal to the Scott topology on (X, ≤).Every directed set D of X under specialization order can be regarded as a monotone net, we say D converges to x iff for every open neighborhood U of x, D U = ∅.We say that V is a directed open set of X if for all directed set D which converges to some point of V , then D V = ∅.It is easy to see that every directed open set is an upper set.
Definition 2.1 [12] Let X be a T 0 space.If every directed open set of X is also an open set, then we say that X is a directed space.
There are many important spaces in domain theory which also are directed spaces.Example 2.2 (i) Every poset with Scott topology is a directed space.(ii) All c-spaces are directed spaces.In particular, every Alexandroff space is a directed space.(iii) All locally finitary compact spaces are directed spaces.By the way, every c-space is locally finitary compact.
Next we introduce the concept of the exponential object in general category.
Definition 2.3 Given two objects X, Y in a category C with binary products, an exponential object, if it exists, is an object Y X with a morphism App :

App
The following result describes the underlying set of the exponential object in Top.
Proposition 2.4 [9] Let C be any full subcategory of Top with finite products, and assume that 1 = {⋆} is an object of C. Let X, Y be two objects of C that have an exponential object Y X in C.
Then there is a unique homeomorphism θ : Remark: By the above result, we always let the exponential object in C be the set [X → Y ] with some unique topology if it exists.Theorem 2.5 [12] The category of directed spaces and continuous maps is cartesian closed.
Next, we build a relationship between directed spaces and Scott spaces, which will be used later.Definition 2.6 Let X be a T 0 space.If X with the specialization order is a dcpo and every open set of X is Scott open in (X, ≤).Then we say that X is a d-space.Lemma 2.7 A directed space is a Scott space iff it is a d-space.
Proof.We only need to show the "if" part.Let X be a d-space and a directed space, obviously every open set of X is Scott open of (X, ≤) since X is a d-space.Now take any Scott open set U of (X, ≤) and Since X is a directed space, the topology on X is exactly the Scott topology on (X, ≤).✷ We list some results about separate continuity and joint continuity.
Theorem 2.8 [11] Let E be a T 0 space.The following conditions are equivalent: (i) E is locally finitary compact.
(ii) For all T 0 space X, if a map from X × E is separately continuous, then it is jointly continuous.
Corollary 2.9 Let X be a c-space and Y a T 0 space.For any T 0 space Z, a map f : X × Y → Z is continuous (i.e.jointly continuous) iff it is separately continuous.

The category of c-spaces
We now prove our main result.
Theorem 3.1 The category of c-spaces with continuous maps (CS for short) is not cartesian closed.
Proof.Let Z − be the set of non-positive integers with the Scott topology.Assume CS is a ccc.It is easy to see that the topological product X × Y is the categorical product because X × Y is a c-space.Since CS is cartesian closed, there exists exponential topology τ on [Z − → Z − ], we denote by [Z − → Z − ] τ .Then for any c-space Y and any map f : The specialization order on [Z − → Z − ] τ is equal to the pointwise order.For any for any x ∈ X.For any We only need to show that every directed subfamily (g i ) i∈I of [Z − → Z − ] τ converges to its supremum g = ↑ i∈I g i .Let Y be the set I ∪ {∞} with the topology generated by {↑i ∪ {∞} : It is easy to see that f is continuous since f is continuous iff it is separately continuous by 2.9.It follows that f : Y → [Z − → Z − ] τ is continuous, and so (g i = f (i)) i converges to f (∞) = g.
Therefore τ is just the Scott topology on [Z − → Z − ].But from [10] we know that [Z − → Z − ] is not a continuous domain, hence it is not a c-space, a contradiction.✷ Theorem 3.2 [8] A meet continuous dcpo is a continuous dcpo iff it is a quasicontinuous dcpo.
Notice that [Z − → Z − ] is a meet continuous semilattice which is not continuous, hence it is not a quasicontinuous dcpo.Then we have the following result.

Corollary 3.3
The category of locally finitary compact spaces with continuous maps is not cartesian closed.
set D converges to some point x of U .Assume that D U = ∅, then b = ↑ D ∈ U .It follows that x ∈ X\↓b.Because D converges to x and X\↓b is open in X, there is some d ∈ D such that d ∈ X\↓b,, a contradiction.Hence the assumption is wrong.It means that U is a directed open set of X.