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Scientific work
The first of my scientific work published in
1949 had been written as diploma paper in 1947, i.e., more than fifty
years ago. This paper was devoted to investigation of some special properties
of poiihedra. The subsequent work dealt with geometric topology too,
namely with non-trivial imbeddings of zero-dimensional compacts into
euclidean spaces (generalization of Antoin's example). Several years
I took up different applications .
I like to search new ideas having no analogue,
to create on their basis original scientific theories with minimal restrictions
concerning basic notions. Such approach allows us to understand the
corresponding significance of either restriction. For example, the extension
theory had been constructed for arbitrary topological spaces with no
separation axiom, the same approach took place for extension structures.
Here the following structures are meant: topological structures , proximity
relations , contiguity relations , uniform structures . These directions
of investigations always were attracting and are attracting my attention.
Here new theories were created.
The fixed point theory was also
very interesting for me, especially the fixed point theory of metric
space mappings. There are many publications of different kind , which
I prepared on this subjects. One of such publications, which combines
a surway of the theory with an account of new results, is rather characteristic
for me. I like such construction of papers and often use one. The fact
is that such form of paper allow to make clear the place, which any
new result takes up in the whole theory.
Categorical approach also helps
to understand the main point. All my publications on category theory
had just such purpose, for example the concept of enrichments of categories
unites the categories of topological spaces, proximity space, contiguity
spaces and uniform spaces. Some results on category theory I included
in non-categorical papers. It concerns, of course, the so-called topological
type structures and categories of topological type spaces, first of
all the category of bitopological spaces , the category of settopological
spaces and the category of ratio spaces , only bitopological spaces
having been studied systematically.
There are vast literature devoted
to bitopological spaces. The notion of bitopological space in initial
variant was understood as a set with two topologies given on this set.
The theory of bitopological spaces in such understanding (classical
theory) consists of generalization of the usual theory of topological
spaces on bitopological spaces, i.e., in passing from one topology given
on a set to two topologies --rather trivial approach. The general theory
of bitopological spaces created in my papers deals with perfectly new
notions having no analogue in topological theory. It proved to be that
this theory has interesting applications to topology and algebra. I
mean bitopological representations of classes of continuous mappings
of topological spaces, bitopological manifolds , bitopological groups
.
Lately a very interesting theory
was initiated, namely the so-called duality theory of topologies on
products and ratios, This theory gives us, in particular, a possibility
of joint study of bitopological, settopological and ratio spaces. Now,
I am going to study more systematically the theory of space structures.
As the result of this work a book "Space structures, theory and
applications" will be written.
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