一般拓扑学中的几个问题

【摘要】： 1) If X is a regular 6- refinable space, then X is DC-like space if and only if A' is CSK-like space, thus the problem appeared in [2] is partly answered; 2) If Xn is a regular Lindelof DC-like space for each n W, then fl Xn is a n6JV Lindelof space; 3) A is a //-set of space X if and only if A is closed in Y whenever Y is a T? space, A is contained in an open set U of A' and U is embedded in Y , thus the question appeared in [15] is answered; Y C A, Y is countably //-set if and only if Y is closed in Z whenever Z is a T3 first countable space, Y is contained in an open set V of AT and V is embedded in Z; Y C X, Y is a countably //-bounded set if and only if Y is closed in every T3 first countable space Z in which X is embedded; 4) An example is given to show that the main result of [21] and 3.2.14 of [23] should be considered again; 5) Let X be a TI, S'-space and every point of X is (-set, if / : X - Y is a closed onto map, then there is a cr-discrete subspace Z of V, such that f~l(y) is an ui -compact subset of X for every y € Y \ Z\ 6) If X is a regular fc-space with a a- HCP closed fc-network, then X is a hereditarily metalindelof space, thus the question appeared in [29] and [30] is answered; A normal sequential space with a a-HCP fc-network is a paracompact space; 7) If X is a regular space, and X = U{.Yn : n ?N}, Xn has a r-locally finite PF-regular closed net, Xn is a closed subset of X for each n € N, then X has a cr-locally finite PF-regular closed net; If X is a Lasnev C-scattered space, then X is a //F-netted space; 8) Two problems and some conclusions about relative topology are discussed.