Abstract
It is well known that every epireflective, full subcategory of the category Comp 2 of compact Hausdorff spaces is algebraic in the sense of HERRLICH [6,§32]. Conversely, every algebraic, epireflective, full subcategory of the category of all Hausdorff spaces is contained in Comp 2. This generalizes a result of HERRLICH and STRECKER [5] and yields a complete new proof for it. The lattice of such algebraic categories is very large.
For arbitrary full subcategories C of topological (not necessary Hausdorff) spaces the following holds:
If C is algebraic, closed-hereditary, and contains the ordinal spaces [O,β] for every limit ordinal β then each space in C is compact (not necessary Hausdorff).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
P. Alexandroff, P. Urysohn, Zur Theorie der topologischen Räume, Math. Ann. 92 (1924) 258–266
B. Banaschewski, H. Herrlich, Subcategories defined by implications, Houston J. of Math. 2 (1976) 149–171
R.H. Bing, A homogenous indecomposable plane continuum, Duke Math. J. 15 (1948) 729–741
H. Herrlich, Topologische Reflexionen und Coreflexionen, Springer Lecture Notes in Math. 78 (1968)
H. Herrlich, G.E. Strecker, Algebra ∩ Topology = Compactness, General Topology and its Applications 1 (1971) 283–287
H. Herrlich, G.E. Strecker, Category Theory, Allyn and Bacon, Boston (1973)
M. Hušek, J. van Mill, Ch. F. Mills, Some very small continua, Topological structures II, Math. Centre Tracts 115 (1979) 147–151
F.E.J. Linton, Some aspects of equational categories, Proc. Conf. Categorical Algebra, La Jolla 1965 (1966) 84–94
N. Noble, Products with closed projections II, Trans. Amer. Math. Soc. 160 (1971) 169–183
A. Pultr, V. Trnková, Combinatorial algebraic and topological Representations of Groups, Semigroups and Categories, Noth-Holland Math. Lib. 22 (1980)
G. Richter, Kategorielle Algebra, Studien zur Algebra und ihre Anwendungen 3 (1979)
E. Wattel, The compactness operator in set theory and topology, Math. Centre Tracts 21 (1968)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1982 Springer-Verlag
About this paper
Cite this paper
Richter, G. (1982). Algebraic categories of topological spaces. In: Kamps, K.H., Pumplün, D., Tholen, W. (eds) Category Theory. Lecture Notes in Mathematics, vol 962. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066907
Download citation
DOI: https://doi.org/10.1007/BFb0066907
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-11961-6
Online ISBN: 978-3-540-39550-8
eBook Packages: Springer Book Archive