On the cross topology of the plane
by
Strashimir G. Popvassilev
Auburn University and Institute of Mathematics of the Bulgarian Academy of Sciences

We investigate the cross topology of the plane that is defined as the finest one which induces the standard topology on each vertical line and on each horizontal line. There is a set dense in the usual topology, which meets each line in at most two points, and hence is closed in the cross topology (one constructs such a set starting with a countable base of the usual topology and picking inductively the n-th point of the set in the n-th element of the base). Also, the cross-closure of each cross-open (non-empty) set contains a (non-empty) standard open set (one uses that if an open interval is a union of countably many closed sets, then one of them has non-empty interior). From this it follows that the cross topology is not regular. The character of the cross topology is greater than the cardinality of continuum, though it is separable, which gives another proof of the non-regularuty. These two proofs do not work for the topology obtained by taking as a basis all regular open sets of the cross topology (i.e. sets that equal the interior of their closure), the so called semi-regularization. We present a third proof of the non-regularity of the cross topology, which uses the picture of a self-similar subset of the plane. This proof is easy to describe, and it proves the non-regularuty of the semi-regularization. A proof of the non-regularity of (a very similar to) the cross topology is contained in [1]. Another proof and applications to separate continuity are found in [2]. The author's work is in [3-5]. New results about separate continuity and the cross topology are contained in [6]. Non-standard topologies of Minkowski space-time that induce the standard topology on each space axis and on each time axis find applications in physics [7], [8]. [1] J. Novak, Topologies defined by a class real-valued functions, Gen. Top. Appl., 1(1971), 247-251. [2] C.J. Knight, W. Moran, J.S. Pim, The topologies of separate continuity, I and II, Proc. Camb. Philos. Soc., 68(1970), 663-671 and 71(1972), 307-319. [3] S.G. Popvassilev, Non-regularity of some topologies on R^n stronger than the standard one, Math. Pannon., 5(1994), 1, 105-110. [4] S.G. Popvassilev, Principle A^tau, Principle A^tau, II, Q A in Gen. Topol., 13(1995), 2, 203-205, 207-210. [5] S.G. Popvassilev, Baire property versus non-regularity in some topologies on R^n, C.R. Acad. Bulg. Sci, 49(1996), 5, 11-14. [6] M. Henriksen, R.G. Woods, Separate vs. joint continuity A tale of four topologies, a version (without proofs) was published in the Proc. Tenessee Topol. Conf., World Sci., 1997, 67-84. [7] E.C. Zeeman, The topology of Minkowski space, Topology, 6(1967), 161-170. [8] O. Laback, On the timelike relation in space-time, Acta Phys. Austriaca, 52(1980), 293-299.

Date received: February 13, 1998