A Note On Function Space And 1-Sequence-Quotient Mapping

This paper is presented as three sections. In the first section it is proved that Weierstrass's approximation theorem in read analysis is hold in function spaces endowed with set-open topology. In the second section we mainly discusses Ascoli theorem and prove that if there is some compact k-network under finite unions in function spaces Ascoli theorem is hold in set-open topology too. In the three section we show that a space has a point-countable weak base if and only if it is a 1-sequence-quotient, quotient and s-image of a metric space. Particularly, we prove that K-spaces are preserved by 1-sequence-quotient, closed mappings which answers a problem posed by Gu Jiansheng.