Corrections to \ Some Generalizations of Td - Spaces " and \ a Generalization of Normal Spaces

Some corrections to the papers \Some Generalizations of TD-Spaces (Mat. Ves-nik 34 (1982), 221{230)" and \A Generalization of Normal Spaces (ibid. 35 (1983), 1{10)" are given. 1. Semi-T D spaces 1] The rst part of the proofs of Theorems 1.6 and 2.4 of 1] are incorrect. The following two Theorems and their proofs provide the statements and proofs of the rst parts of the Theorems 1.6 and 2.4 of 1]. The proofs of Theorems 1:6 and 2:4 1] given in the paper prove the second parts of the statements of the Theorems. Theorem 1.1. A semi-T 1 space is semi-T D. Proof. Let X be a semi-T 1 space and x 2 X. Then either fxg is nowhere dense or fxg int(clfxg) 3] where int A and cl A respectively denote the interior and closure of a set A. If fxg is nowhere dense, cl(X ? clfxg) = X and therefore, X ?clfxg X ?dfxg cl(X ?clfxg). So, X ?dfxg is semi-open, dfxg being the derived set of fxg. If fxg is not nowhere dense, fxg int(clfxg) = sclfxg, the semi-closure of fxg 3]. Since X is semi-T 1 , fxg = int(clfxg). Hence dfxg = clfxg?fxg, the boundary of an open set and hence a nowhere dense set. Therfore X ? dfxg is semi-open. Thus dfxg is semi-closed. Theorem 1.2. A pairwise semi-T 1 space is pairwise semi-T D. T 2) are semi-T 1 and hence both are semi-T D , in view of the above Theorem. For x 2 X, T 1-dfxg is T 1-semi-closed and T 2-dfxg is T 2-semi-closed. Suppose y = 2 (T 1-dfxg\T 2-dfxg). If y = 2 (T 1-dfxg T 2-dfxg), then there is a T 1-semi-open set U and a T 2-semi-open set V , each containing y, and each having empty intersection with (T 1-dfxg\T 2-dfxg). Now suppose y 2 (T 1-dfxggT 2-dfxg) and y = 2 (T 1-dfxg\T 2-dfxg). Suppose y 2 T 1-dfxg and y = 2 T 2-dfxg. Then there is a T 2-semi-open set