V. Kannan

Constructions and applications of rigid spaces, I

Abstract The two major results proved are: (1) The category TOP of topological spaces contains a complete nonreflective subcategory. (2) Under the assumption (2 m ) + 2 m , for each infinite cardinal number m there exists a Hausdorff space of cardinality m , in which the identity map is the only nonconstant continuous self-map. The first result is proved as a consequence of another result which answers a question of Herrlich concerning strongly rigid spaces; it is then used to settle in the negative a conjecture concerning the characterization of reflective subcategories in TOP. In addition, several interesting spaces are constructed.

On scattered spaces

We show that each O-dimensional Hausdorff space which is scattered can be mapped continuously in a one-to-one way onto a scattered O-dimensional Hausdorff space of the same weight as its cardinality. This gives an easier and a new proof of the fact that a countable regular space admits a coarser compact Hausdorff topology if and only if it is scattered. We also show that a 0-dimensional, Lindelof, scattered first-countable Hausdorff space admits a if it is scattered. We also show that a 0-dimensional, Lindelof, scattered first-countable Hausdorff space admits a scattered compactification. In particular we give a more direct proof than that of Knaster, Urbanik and Belnov of the fact that a countable scattered metric space is a subspace of [1, Ω), and deduce a result of W. H. Young as a corollary.

Properties that are productive, closed-hereditary and surjective

We obtain a characterization of all those topological properties of regular Hausdorff spaces, that are preserved under the formation of arbitrary products, closed subspaces and continuous surjections.

A countable connected Urysohn space containing a dispersion point

Answering a question of Martin, Roy gave the first example of a countable connected Urysohn space with a dispersion point. Here we give a much simpler example of such a space.

Simultaneous best approximation by three polynomials

Abstract Let [ a , b ] be any interval and let p 0 , p 1 , p k be any three polynomials of degrees 0, 1, k , respectively, where k ⩾ 2. A set of necessary and sufficient conditions for the existence of an f in C [ a , b ] such that p i is the best approximation to f from the space of all polynomials of degree less than or equal to i , for all i = 0, 1, k , is given.

Spaces every quotient of which is metrizable

We characterise those topological spaces for which every quo- tient image is metrizable. This supplements the earlier known results in this direction, in a fairly complete manner.