Topology Atlas Document # taic-34 Topology Atlas Invited Contributions vol. 5 issue 1 (2000) 1-4

## Fibrewise General Topology: A Brief Outlook

### Invited Contribution

The study of General Topology is usually concerned with the category TOP of topological spaces as objects, and continuous maps as morphisms. The concepts of space and map are equally important and one can even look at a space as a map from this space onto a singleton space and in this manner identify these two concepts. With this in mind, a branch of General Topology which has become known as General Topology of Continuous Maps, or Fibrewise General Topology, was initiated. This field of research is concerned most of all in extending the main notions and results concerning topological spaces to those of continuous maps. In this way one can see some well-known results in a new and clearer light and one can also be led to further developments which otherwise would not have suggested themselves. The fibrewise viewpoint is standard in the theory of fibre bundles, however, it has been recognized relatively recently that the same viewpoint is also as important in other areas such as General Topology.

For an arbitrary topological space Y one considers the category TOPY, the objects of which are continuous maps into the space Y, and for the objects f : X --> Y and g : Z --> Y, a morphism from f into g is a continuous map \lambda : X --> Z with the property f = g \circ \lambda. This situation is a generalization of the category TOP, since the category TOP is isomorphic to the particular case of TOPY in which the space Y is a singleton space.

The carried out research showed a strong analogy in the behaviour of spaces and maps and it was possible to extend the main notions and results of spaces to that of maps. Since the considered case is of a wider generality (compared to that of spaces), the results obtained for maps are technically more complicated. Moreover there are moments which are specific to maps. For example, there is no analogue to Urysohn's Lemma for maps and so normality and functional normality do not coincide and as a consequence, there exist two theories of compactifications, one for Hausdorff compactifications and one for Tychonoff compactifications. One can consult [20, 5, 1] for details on compactifications of maps.

Some results in the General Topology of Continuous Maps were obtained quite some time ago. For example, in 1947, I.A.Vainstein [23] proposed the name of compact maps to perfect maps, G.T.Whyburn in 1953 [24,25], as did G.L.Cain, N.Krolevets, V.M.Ulyanov [22] and others, considered compactifications of maps. In the meantime, until quite recently, there wasn't a connected unified theory for maps. One of the main reasons might have been the lack of separation axioms for maps, especially that of Tychonoffness (and complete regularity) and also that of (functional) normality and collectionwise normality.

Completely regular and Tychonoff maps, as well as (functionally) normal maps, were defined by B.A.Pasynkov in 1984 [19]. These definitions made it possible to generalize and obtain an analogue to the theorem on the embedding of Tychonoff spaces of weight \tau into I\tau and to the existence of a compactification for a Tychonoff space having the same weight. It was also possible to construct a maximal Tychonoff compactification for a Tychonoff map (i.e. construct an analogue to the Stone-Cech compactification). Collectionwise normal maps were defined by the author [8] and enabled the definition of metrizable type maps, giving a satisfactory fibrewise version of the theory of metrizable spaces.

In [2, 3], a category of maps MAP in which one does not restrain oneself with a fixed base space Y was studied. The objects of MAP are continuous maps from any topological space into any topological space. For two objects f1 : X1 --> Y1 and f2 : X2 --> Y2, a morphism from f1 into f2 is a pair of continuous maps {\lambdaT, \lambdaB}, where \lambdaT : X1 --> X2 and \lambdaB : Y1 --> Y2, such that \lambdaB \circ f1 = f2 \circ \lambdaT It is not difficult to see that this definition of a morphism in MAP satisfies the necessary axioms that morphisms should satisfy in any category (see, for example, [21]). One can note that when Y1 = Y2 and \lambdaB = idB, we have the definition of morphism in TOPY.

Let PT and PB be two topological/set-theoretic properties of maps (for example: closed, open, 1-1, onto, etc.). If \lambdaT has property PT and \lambdaB has property PB then {\lambdaT, \lambdaB} is called a {PT, PB}-morphism. If PT is the continuous property, then {\lambdaT, \lambdaB} is said to be a {*, PB}-morphism, similarly for PB. Therefore, a {*,*}-morphism is just a morphism. Also, if PT = PB = P then a {PT, PB}-morphism is called a P-morphism.

In most cases there is some choice in defining properties on maps and one usually prefers the simplest and the one that gives the most complete generalization of the corresponding results in the category TOP. Needless to say, since any topological property PY on maps (as objects of TOPY or MAP) can be considered as a generalization of a corresponding property P on spaces (as objects of TOP), PY must coincide with P when the base space Y is a singleton space. It would be beneficial to have a more systematic way of extending definitions and results from the category TOP to the category MAP and some hope is provided by the link between Fibrewise Topology and Topos Theory [13, 14, 16, 17]. Unfortunately, as was noted in [12], this approach has several drawbacks. In defining compact maps [20], paracompact maps [4], metacompact maps, subparacompact maps, submetacompact maps [6] and metrizable type maps [8], one can see a systematic method in defining notions in the category TOPY (or more general in the category MAP) corresponding to definitions which involve coverings or bases of topological spaces. This construction gave satisfactory definitions which can be seen from the results obtained for such maps [4, 6, 8, 20]. One can also add that the definitions of paracompact maps, metacompact maps, subparacompact maps and submetacompact maps strengthened the result that paracompactness, metacompactness, subparacompactness and submetacompactness are all inverse invariant of perfect maps. Namely, it was proved that the inverse image of a paracompact T2 (resp. subparacompact, metacompact, submetacompact) space by a paracompact T2 (resp. subparacompact, metacompact, submetacompact) map is paracompact T2 (resp. subparacompact, metacompact, submetacompact) [4, 6]. External characterizations of paracompact maps in the category MAP, where the product of objects is the natural Tychonoff product of maps, were given in [5].

Another possible systematic way of extending definitions from the category TOP to the category TOPY (or even to the category MAP) was given in [7]. This is done in the following way: Let C be some class of topological spaces. A map f : X --> Y is said to be trivially C (or TC) if it is parallel to a space C in C, i.e. there exists a space C in C and an embedding e : X --> Y × C such that f = prY \circ e, where prY : Y × C --> Y is the projection of the product onto the factor Y. A map f is said to be locally trivially C (or LTC) if for any y in Y, there exists a neighbourhood Oy of y such that the restriction f|f-1Oy : f-1Oy --> Oy is a TC-map. One can note that in the definition of LTC-map, the space Cy in C can be different for every f|f-1Oy : f-1Oy --> Oy. Two classes of spaces were considered in [7] as the collection C, the class of all metrizable spaces and the class of all linearly ordered topological spaces (i.e., LOTS). This method gave another possible way of defining a metrizable map.

There is a rich potential for research to be found in FGT, and even more in fibrewise topology in general. In many ways, research in fibrewise topology is still at a pioneering stage. One can add that a lot of research has been done on fibrewise homotopy theory [10, 11, 12, 9] and that a theory of fibrewise absolute (neighbourhood) retracts has been developed in [18]. One can note that in the mentioned recent paper [9] on fibrewise fibrations and cofibrations, one can see the advantages of using the wider category MAP.

### References

[1]
I.V. Bludova and G. Nordo, On the poset of all Hausdorff and all Tychonoff compactifications of continuous mappings, Q & A in General Topology 17 (1999), 47-55.

[2]
D. Buhagiar, The category MAP, submitted for publication.

[3]
..., Partial topological products in MAP, submitted for publication.

[4]
..., Paracompact maps, Q & A in General Topology 15 (1997), no. 2, 203-223.

[5]
..., The Tamano Theorem in MAP, Comment. Math. Univ. Carolinae 40 (1999), no. 4, 755-770.

[6]
D. Buhagiar and T. Miwa, Covering properties on maps, Q & A in General Topology 16 (1998), no. 1, 53-66.

[7]
D. Buhagiar, T. Miwa, and B.A. Pasynkov, Trivially and locally trivially C maps, submitted for publication.

[8]
..., On metrizable type (MT-) maps and spaces, Topology Appl. 96 (1999), 31-51.

[9]
T. Hotta and T. Miwa, A new approach to fibrewise fibrations and cofibrations, submitted for publication.

[10]
I.M. James, Ex-Homotopy theory I, Illinois J. Math. 15 (1971), 324-337.

[11]
..., Alternative homotopy theories, Enseign. Math. 23 (1977), no. 2, 221-247.

[12]
..., Fibrewise topology, Cambridge Univ. Press, Cambridge, 1989.

[13]
P.T. Johnstone, The Gleason cover of a topos II, J. Pure and Appl. Algebra 22 (1981), 229-247.

[14]
..., Wallman compactification of locales, Houston J. Math. 10 (1984), 201-206.

[15]
H.P.A. Künzi and B.A. Pasynkov, Tychonoff compactifications and R-completions of mappings and rings of continuous functions, Categorical Topology (L'Aquila, 1994), Kluwer Acad. Publ., Dordrecht, 1996, pp. 175-201.

[16]
D. Lever, Continuous families: Categorical aspects, Cahiers de topologie et géométrie différentielle 24 (1983), 393-432.

[17]
..., Relative topology, Categorical Topology, Proc. Conference Toledo, Ohio, 1983, Heldermann, Berlin, 1984.

[18]
T. Miwa, On fibrewise retraction and extension, to appear in Houston J. Math.

[19]
B.A. Pasynkov, On extension to mappings of certain notions and assertions concerning spaces, Mappings and Functors (Moscow), Izdat. MGU, Moscow, 1984, in Russian, pp. 72-102.

[20]
..., Elements of the general topology of continuous maps, On Compactness and Completeness Properties of Topological Spaces (Tashkent), "FAN" Acad. of Science of the Uzbek. Rep., Tashkent, 1994, in Russian, pp. 50-120.

[21]
G. Preuss, Theory of topological structures (an approach to categorical topology), D.Reidel Publishing Company, Dordrecht, Holland, 1987.

[22]
V.M. Ulyanov, On compact extensions of countable character and absolutes, Math. Sb. 98 (1975), no. 2, 223-254, in Russian.

[23]
I.A. Vainstein, On closed maps of metric spaces, Doklady Akad. Nauk SSSR 57 (1947), 319-321, in Russian.

[24]
G.T. Whyburn, A unified space for mappings, Trans. Amer. Math. Soc. 74 (1953), no. 2, 344-350.

[25]
..., Compactification of mappings, Math. Ann. 166 (1966), no. 1, 168-174.

David Buhagiar
Mathematics Department
Faculty of Science
University of Malta
Msida MSD.06, Malta
dbuha@maths.um.edu.mt

Date received: June 2, 2000.