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In algebra the concepts of a finitely generated and finitely presentable object work well: an algebra A in a variety has finitely many generators iff it is a finitely generated object (the hom-functor of A preserves directed unions). And A is presented by finitely many generators and relations iff it is finitely presentable (its hom-functor preserves directed colimits).
In topology these concepts are less useful: Gabriel and Ulmer [1] proved that a topological space is finitely presentable in Top iff it is finite and discrete. In subcategories such as Haus (of Hausdorff spaces) or Top1 (of T1 spaces) the situation appears to be even worse: the only finitely presentable object is the empty space. However, unlike algebra, topology offers many interesting subclasses M of that of all monomorphisms, e.g. all embeddings or open embeddings. A space is called finitely generated with respect to M if its hom-functor preserves directed colimits with connecting maps in M.
Theorem. In Top, Top1 or Haus a space is finitely generated with respect to open embeddings iff it is compact.
Thus in a sense 'small spaces' are just the compact ones.
We present a number of related results, e.g. those applying the concept of finitely small space A as used in algebraic topology [2]: the hom-functor of A preserves colimits of smooth chains. Although chains and directed colimits are usually equivalent, in the category Top1 a space is
(1) Finitely generated with respect to embeddings iff it is finite.
(2) Finitely small with respect to embeddings iff it is countable and compact.
These results are available in arXiv [3].
References:
[1] P. Gabriel and F. Ulmer: Lokal präsentiebaren Kategorien. Lecture Notes in Mathematics 221, Springer-Verlag 1971.
[2] D.G. Quillen: Homotopical Algebra, Lecture Notes in Mathematics 43, Springer-Verlag, 1967.
[3] J. Adámek, M. Husek, J. Rosicky and W. Tholen: Smallness in topology, accepted for publication, arxiv.org/pdf/2302.00050.
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