Carlos R. Borges

Professor Emeritus
General topology
Ph.D., 1968, California Institute of Technology
Refereed publications: Via Math Reviews

Email: cborges@math.ucdavis.edu

Professor Carlos Borges' research is in the area of general topology, the study of highly abstracted topological spaces. In particular, he is interested in stratifiable spaces and other spaces which have some but not all properties of metrizable spaces.

The most common objects in topology, such as manifolds, polyhedra, and Euclidean spaces, are metrizable spaces, which means that one can describe the topology by a (real-valued) distance between points, and say that a sequence of points converges to a limit if and only if the distance decreases to zero. However, many of the ideas of topology are still useful when one considers spaces have the bare minimum structure needed to define continuity. Such a structure is called a topology or a topology of open sets. The study of these general spaces is called general topology and it has connections to analysis and logic.

In several papers [1] [3], Professor Borges developed the notion of stratifiable spaces, which includes many relatively tame spaces which are too big to be metrizable. He showed that a number of basic structure theorems about metrizable spaces, for example that the quotient of a first-countable metrizable space is metrizable, holds for stratifiable spaces also.

Professor Borges has also considered questions of existence of extensions of continuous functions [2] [4] [5]. For example, he proved that a locally convex vector space X is an extensor space for stratifiable spaces, meaning that if Y is a stratifiable space and Z is a closed subspace, a continuous function from Y to X can be extended to Z. The same is true if Z is a closed, convex subset of a locally convex linear space. This result extends the Dugundji extension theorem, which in turn is a stronger version of the Tietze extension theorem.

Selected publications

[1] On stratifiable spaces. Pacific J. Math. 17 (1966), 1-16.

[2] A study of absolute extensor spaces, Pacific J. Math. 31 (1969) 609-617.

[3] Sup-characterization of stratifiable spaces (with G. Gruenhage), Pacific J. Math. 105 (1983) 279-284.

[4] Retraction properties of hyperspaces, Math. Japonica 30 (1985), 551-557.

[5] Hyperconnectivity of hyperspaces, Math. Japonica 30 (1985), 757-761.

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