Data di Pubblicazione:
2013
Citazione:
Topologies and free constructions / A. Bucalo, G. Rosolini. - In: LOGIC AND LOGICAL PHILOSOPHY. - ISSN 1425-3305. - 22:3(2013 Sep), pp. 327-346. [10.12775/LLP.2013.015]
Abstract:
The standard presentation of topological spaces rely heavily on (na ve) set theory: a
topology consists of a set of subsets of a set (of points). And many of the high-level tools of
set theory are required to achieve just the basic results about topological spaces.
Concentrating on the mathematical structures, category theory o ers the possibility to
look synthetically at the structure of continuous transformations between topological spaces
addressing speci cally how the fundamental notions of point and open come about. As a
byproduct of this, one may look at the di erent approaches to topology from an external
perspective and compare them in a uni ed way.
Technically, the category of sober topological spaces can be seen as consisting of (co)algebraic
structures in the exact completion of the elementary category of sets and relations. Moreover,
the same abstract construction of taking the exact completion, when applied to the category
of topological spaces and continuous functions produces an extension of it which is cartesian
closed. In other words, there is one general mathematical construction that, when applied
to a very elementary category, generates the category of topological spaces and continuous
functions, and when applied to that category produces a very suitable category where to
deal with all sorts functions spaces.
Yet, via such free constructions it is possible to give a new meaning to Marshall Stone's
dictum: \always topologize" as the category of sets and relations is the most natural way
to give structure to logic and the category of topological spaces and continuous functions is
obtained from it by a good mix of free constructions.
Tipologia IRIS:
01 - Articolo su periodico
Keywords:
Topological spaces
Elenco autori:
A. Bucalo, R. G.
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