# A New Category? Domains, Spaces and Equivalence
Relations

The familiar categories **SET** and **TOP**, consisting of sets
and arbitrary mappings and of topological spaces and continuous
mappings, have many well known closure properties. For example, they
are both complete and cocomplete, meaning that they have all (small)
limits and colimits. They are well-powered and co-well-powered,
meaning that collections of subobjects and quotients of objects can be
represented by sets. They are also nicely related, since **SET**
can be regarded as a full subcategory of **TOP**, and the forgetful
functor that takes a topological space to its underlying set preserves
limits and colimits (but reflects neither). The category **SET** is
also a *cartesian closed category*, meaning that the
*function-space construct* or the *internal hom-functor* is
very well behaved. However, it has been known for a long time that in
**TOP** no such assertion is available, because in general it is
not possible to assign a topology to the set of continuous functions
*A*^{B} making this adjointness valid -- except under
some special conditions on the space *B*.
The proposed solution to the problem of cartesian closedness is
motivated by domain theory. The (new?) category is the the category of
topological *T*_{0}-spaces and arbitrary equivalence
relations, to be called **EQU**, where the mappings are (suitable
equivalence classes of) continuous mappings which preserve the
equivalence relations. Let us call these spaces **equilogical
spaces** and the mappings **equivariant**. It seems surprising
that this category has not been noticed before -- if in fact it has
not. It is easy to see that **EQU** is complete and cocomplete and
that it embedds **TOP**_{0} as a full and faithful
subcategory (by taking the equivalence relation to be the identity
relation). What is perhaps not so obvious is that **EQU** is indeed
cartesian closed. The proof of cartesian closedness uses old theorems
in domain theory originally discovered by the author: in particular,
an injective property of algebraic lattices treated as topological
spaces and the fact that they form a cartesian closed category (along
with continuous functions). The paper discusses other closure
properties of **EQU** as well.

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Table of Contents

**1. Introduction**
**2. ***T*_{0}-Spaces and Algebraic Lattices
**3. Equilogical Spaces**
**4. Some Subcategories**
**5. Acknowledgments and Questions**