BEBERAPA TOPOLOGI PADA RUANG TERINTEGRAL HENSTOCK-KURZWEIL; SOME TOPOLOGIES ON THE HENSTOCK-KURZWEIL INTEGRABLE SPACE

HARYADI, Soeparna Darmawijaya

2011 | Disertasi | PROGRAM STUDI S2 MATEMATIKA

As we know, the Henstock-Kurzweil integrable spaceHK[a; b] has a norm defined by kfk = supfj(HK) R x a fj : x 2 [a; b]g, where f = g defined as f(x) = g(x) almost every x 2 [a; b]. With the norm, HK[a; b] is not complete. In this research we introduce two topologies on the linear space HK[a; b]. First, we defined interval on HK[a; b] and then a topology is constructed by the intervals. Although the topology is seemed to be natural, there are some shortcomings in the topology, i.e. a sequence of function in HK[a; b] that converges uniformly does not implies the sequence converges with respect to the topology. Secondly, we construc a topology by defining a neighborhood system of null function as the collection of all f 2 HK[a; b] such that f(x) 2 U for almost all x 2 [a; b], where U is a neighborhood of zero in R. With the topology, the convergence in HK[a; b] is equivalent to the uniformly convergence almost everywhere, and every Cauchy sequence in HK[a; b] is converges to an element of HK[a; b]. Furthermore, HK[a; b] is not a linear topological space. In addition, the collection of all f 2 HK[a; b] that is bounded almost everywhere on [a; b] is a closed linear topological subspace of HK[a; b].

Kata Kunci : Topologi