A Cartesian closed topological category of sequential spaces. (English) Zbl 0724.18007
The category of sequential spaces SEQ is introduced. Let X be a set and \(\pi\) a mapping which to every point \(x\in X\) assigns a set \(\pi\) (x) of sequences of points of X. The pair (X,\(\pi\)) is called a sequential space provided the following two axioms are satisfied:
(i) If \(x\in X\) and \(x_ i=x\) for all \(i\in {\mathbb{N}}\), then \(\{x_ i\}_{i\in {\mathbb{N}}}\in \pi (x).\)
(ii) If \(\{\{x_{ij}\}_{j\in {\mathbb{N}}^ 0}\}_{i\in {\mathbb{N}}^ 0}\) is a sequence of sequences of points of X such that \(\{x_{ij}\}_{j\in {\mathbb{N}}}\in \pi (x_{i0})\) for all \(i\in {\mathbb{N}}^ 0\) and \(\{x_{ij}\}_{i\in {\mathbb{N}}}\in \pi (x_{0j})\) for all \(j\in {\mathbb{N}}^ 0\), then \(\{x_{ii}\}_{i\in {\mathbb{N}}}\in \pi (x_{00}).\)
SEQ is the category of all sequential spaces with morphisms defined by: Given two sequential spaces \(A=(X,\pi)\) and \(B=(Y,\rho)\) then a mapping \(f:X\to Y\) is a morphism from A to B provided that \(\{x_ i\}_{i\in {\mathbb{N}}}\in \pi (x)\) implies \(\{f(x_ i)\}_{i\in {\mathbb{N}}}\in \rho (f(x))\) for every \(x\in X.\)
It is proved that the category SEQ is topological and cartesian closed. Finally, subcategories of the category SEQ are investigated.
(i) If \(x\in X\) and \(x_ i=x\) for all \(i\in {\mathbb{N}}\), then \(\{x_ i\}_{i\in {\mathbb{N}}}\in \pi (x).\)
(ii) If \(\{\{x_{ij}\}_{j\in {\mathbb{N}}^ 0}\}_{i\in {\mathbb{N}}^ 0}\) is a sequence of sequences of points of X such that \(\{x_{ij}\}_{j\in {\mathbb{N}}}\in \pi (x_{i0})\) for all \(i\in {\mathbb{N}}^ 0\) and \(\{x_{ij}\}_{i\in {\mathbb{N}}}\in \pi (x_{0j})\) for all \(j\in {\mathbb{N}}^ 0\), then \(\{x_{ii}\}_{i\in {\mathbb{N}}}\in \pi (x_{00}).\)
SEQ is the category of all sequential spaces with morphisms defined by: Given two sequential spaces \(A=(X,\pi)\) and \(B=(Y,\rho)\) then a mapping \(f:X\to Y\) is a morphism from A to B provided that \(\{x_ i\}_{i\in {\mathbb{N}}}\in \pi (x)\) implies \(\{f(x_ i)\}_{i\in {\mathbb{N}}}\in \rho (f(x))\) for every \(x\in X.\)
It is proved that the category SEQ is topological and cartesian closed. Finally, subcategories of the category SEQ are investigated.
Reviewer: M.Demlová (Praha)
MSC:
| 18D15 | Closed categories (closed monoidal and Cartesian closed categories, etc.) |
| 54D55 | Sequential spaces |
| 18B30 | Categories of topological spaces and continuous mappings (MSC2010) |
References:
| [1] | E.Čech,Topological Spaces (Revised by Z. Frolík and M. Katětov), Academia, Prague, 1966.MR 35, 2254 |
| [2] | S. Eilenberg andG. M. Kelly, Closed categories,Proc. Conf. Cat. Alg., La Jolla 1965, 421–562.MR 37, 1432 |
| [3] | H.Herrlich, Cartesian closed topological categories,Math. Coll. Univ. Cape Town 9 (1974), 1–16.MR 57, 408 · Zbl 0318.18011 |
| [4] | H.Herrlich and G. E.Strecker,Category theory, 2nd ed., Heldermann Verlag, Berlin, 1979.MR 81e: 18001 · Zbl 0437.18001 |
| [5] | F.Schwarz, Cartesian closedness, exponentiality, and final hulls in pseudotopological spaces,Quaestiones Math. 5 (1982/83), 289–304.MR 84d: 18005 · Zbl 0521.54005 · doi:10.1080/16073606.1982.9632270 |
| [6] | O.Wyler, Function spaces in topological categories,Lecture Notes Math. 719 (1979), 411–420.MR 82m: 54004 · Zbl 0408.18003 · doi:10.1007/BFb0065291 |
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