Download Topological Structures Horst Herrlich

Proceedings of the International Congress of Mathematicians
Vancouver, 1974
Topological Structures
Horst Herrlich
"New points of view on old subjects are needed. There are too many point set topologists, and not enough who pay attention to what a topological space ought to be."
Saunders MacLane, Notices Amer. Math. Soc. 21 (1974), 183.
The concept of a topological space has been a prime object of topological
investigations. Unfortunately it suffers from certain deficiencies such as :
(a) The category Top of topological spaces and continuous maps is not as well
behaved as one would like it to be; e.g., Top is not cartesian closed, i.e., it is not
possible to supply for any pair {X9 Y) of topological spaces the set XY of all continuous maps from Y to X with a topology such that {XY)Z is naturally isomorphic to
XY*Z. Also, in Top the product of quotient maps in general is no longer a quotient
map.
(b) Several important concepts of a topological nature—such as uniform convergence, uniform continuity, and completeness—cannot be expressed in the
framework of the theory of topological spaces.
There have been serious efforts by prominent mathematicians to remedy this
situation. But none of the solutions offered is free from all the deficiencies mentioned above. The purpose of this note is to stimulate discussion on these matters
among point set topologists.
A. General features of convenient topological categories. To remedy the deficiencies mentioned in (a) several substitutes for Top have been suggested, e.g.,
{a) suitable subcategories of Top, e.g.,
the category of Kelley spaces which is the coreflective hull of all compact
Hausdorff spaces in the category Haus of Hausdorff spaces and continuous maps;
the category of compactly generated spaces which is the coreflective hull of all
© 1975, Canadian Mathematical Congress
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compact Hausdorff spaces in Top;
the category of sequential spaces which is the coreflective hull of all (compact)
metrizable spaces in Top.
(/3) suitable supercategories of Top, e.g.,
the category of quasi-topological spaces and continuous maps;
the category of limit spaces and continuous maps;
the category offilter-generatedmerotopic spaces and merotopic maps.
Several of the above categories have some other deficiencies, e.g., we do not
know how to describe them in a sufficiently elegant manner directly from suitable
axioms. But the general features of convenient topological categories A become
apparent:
A should be a cartesian closed concrete category with small fibres, just one
structure on any one-point set, and such that initial structures for arbitrary sources
exist. Equivalenti^, the following conditions should be satisfied :
(Top 1) A is (complete and) cocomplete,
(Top 2) A is (well-powered and) co-well-powered,
(Top 3) for any A-object A the functor Ax- : A -» A preserves colimits,
(Top 4) the terminal A-object T is a separator and the functor hom(7\ —) : A ->
Set preserves colimits.
A list of references concerning this subject is provided in H. Herrlich, Cartesian
closed topological categories, Math. Colloq. Univ. Cape Town 9 (1974), 1-16.
B. Concrete topological structures. Many solutions have been offered for problem
(b), e.g., (generalized) uniform structures, proximity structures, contiguity structures, syntopogeneous structures and merotopic structures. Recent investigations
have shown that suitable axiomatizations of each of the following concepts give
rise to equivalent !) satisfactory solutions of problem (b):
(a) the collection p of all "uniform covers" of a set X,
(b) the collection y of all collections of subsets of X which "contain arbitrary
small members",
(c) the collection f of all collections of subsets of X which are "near" to some
"spot" inside or outside X.
Consider the following axioms on [x :
(NI) fi is a nonempty collection of covers of X.
(N2) A cover of X belongs to p, if it is refined by some member of p.
(N3)0^.
A pair {X9 pi) may be called a preuniform space provided it satisfies the above
axioms. A map/:(Z, p) -> (7, v) may be called uniformly continuous provided the
preimage of each member of v belongs to p,. The category P-Unif of preuniform
spaces and uniformly continuous maps satisfies (Top 1), (Top 2), (Top 4) but not
(Top 3). The functor Ax-\ P-Unif —> P-Unif preserves coequalizers but in general
not coproducts. Most of the above-mentioned "convenient" topological categories
can be nicely embedded into P-Unif.
Before we define several familiar full subcategories of P-Unif by adding further
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axioms, we define the structure y = mer(^) and the structure £ = near(^) associated
with a preuniform structure fi on X:
WeroVVleiilUeVllAe%AczU9
%eÇo{X-B\Beti)1:{ji.
Then each of the structures y9 respectively £, contains complete information about
fi since
tti/io{Jir-l/|l/eU}^eoV«6r3i4G«3ÜGU,i4c:J7.
Further axioms :
( N 4 ) U e / i and B e ^ imply { u p K | i 7 e t t a n d K e S j e ^
(N5) Vie/i implies {int,, U\ Ue U} e fi where * G int^(C/) o {17, Z — {x}} G fi.
(N6)tte// implies {Fez Z | 3 Î7 G U { £/, X - K j e ^ e / i .
(N7) Every U G ^ has a star-refinement in //.
(N8) Every Vie fi is refined by some finite member of fi.
(N9) If {ini, U\UBVL] covers X then U G fi.
Some results. The full subcategory of P-Unif whose objects satisfy axiom (N4) is
denoted by Q-Unif. It is bicoreflective in P-Unif and isomorphic to each of the
following categories :
(a) the category of quasi-uniform spaces and uniformly continuous maps in the
sense of J. R. Isbell,
(b) the category of merotopic spaces and merotopic maps in the sense of M.
Katëtov,
(c) the category of discrete structure spaces in the sense of D. Harris,
(d) the category of quasi-nearness spaces and nearness preserving maps in the
sense of H. Herrlich.
(N4) and (N5) is denoted by Near. It is bireflective in Q-Unif.
(N4) and (N6) is denoted by R-Near. It is a bireflective subcategory of Near and
equivalent to the category of regular T-uniform spaces in the sense of K. Morita,
resp. to the category of semiuniform spaces in the sense of A. K. Steiner and E. F.
Steiner.
(N4) and (N7) is denoted by Unif. It is a bireflective subcategory of Near and
isomorphic to the category of uniform spaces in the sense of A. Weil and J. W.
Tukey.
(N4), (N5) and (N8) is denoted by Cont. It is a bireflective subcategory of Near
and isomorphic to the category of contiguity spaces in the sense of V. M. Ivanova,
A. A. Ivanov, and W. L. Terwilliger.
(N4), (N5) and (N9) is denoted by Top. It is a bicoreflective subcategory of Near
and isomorphic to the category of symmetric ( = RQ-9 — weakly regular) topological spaces and continuous maps.
(N4), (N6) and (N8) is denoted by Prox. It is bireflective in Near and isomorphic
to the category of proximity spaces and 5-continuous maps in the sense of V. A.
Efremovic and Yu. M. Smirnov.
Furthermore, if an object of Near is called complete provided each (nonempty)
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maximal dement of £ contains an adherence point, then there is a canonical construction by means of which any Near-object can be embedded densely into a
complete Near-object. This completion generalizes the uniform completion of
uniform spaces and the various distinguished Hausdorff compactifications of
suitable topological, uniform, and proximity spaces. Moreover any strict extension,
hence especially any regular extension, of an arbitrary topological 7\-space can be
obtained in this way.
Further, a topological i?0-space is compact iff—considered as a Near-object—it is
contigual, i.e., iff it satisfies (N8). A topological ify-space is fully normal (= regular
paracompact) iff—considered as a Near-object—it is a uniform space, i.e., iff it
satisfies (N7).
A topological i?0-space is regular iff—considered as Near-object—it is regular,
i.e., iff it satisfies (N6).
A topological i?0-space is normal iff—considered as Near-object—its contigual
reflection is regular (equivalently, uniform).
Further details and a reference list are contained in H. Herrlich, Topological
structures, Math. Centrum Amsterdam 52 (1974), 59-1^.
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