Imanges Of Locally Compact Lindel(?)f Spaces And Other Results

This paper is composed of two parts. In the first part, We study spaces with k-system and present some characterizations of spaces with k-system to be the images of locally compact Lindelof spaces by means of quotient mappings, closed mappings or compact-covering mappings. The second part discusses the effects of 1-sequence-quotient mapping and 2-sequence-quotient mapping. The main results are as following:Result 1(Theorem 2.6) For a space X, the following conditions are equivalent: ( 1 ) X is the compact-covering, countable-to-one, SL image of locally compact Lindelof space( 2 ) X is the compact-covering image of locally compact Lindelof space ( 3 ) X has a countable k-cover consisting of compact subsets Result 2(Theorem 2.11) For a space X, the following conditions are equivalent: ( 1 ) X is the compact-covering, finite-to-one, closed, SL image of locally compact Lindelof space( 2 ) X is locally compact Lindelof space ( 3 ) X has a σ-countable and discrete strong k-system ( 4 ) X has a σ-countable and locally finite strong k-system ( 5 ) X has a countable and locally finite k-system( 6 ) X has a countable, point-finite and hereditarily closed-preserving k-system Result 3(Theorem 3.4) If X is sof countable space, then the images of X under 2-sequence-quotient mapping remain the same.Result 4(Theorem 3.5) For a space X, the following conditions are equivalent: (1) X is the 1-sequence-covering mapping ( 2-sequence-covering mapping ) image of metric space(2) X is the 1-sequence-quotient mapping ( 2-sequence-quotient mapping )...