NETS IN TOPOLOGICAL SPACES – INADEQUACY OF SEQUENCES to ADEQUACY OF NETS
NETS IN TOPOLOGICAL SPACES – INADEQUACY OF SEQUENCES to ADEQUACY OF NETS M. VELRAJAN In a metric space, sequences are adequate for closed sets, continuous functions and compact spaces. i.e. in a metric space closed sets, continuous functions and compact spaces are determined by sequences. In fact, 1. Suppose A is a nonempty subset of a metric space X and A be the closure of A in X. Then x ∈ A if and only if there exists a sequence (a n ) in A such that a n → x in X. 2. Suppose (X, d X ) and (Y, d Y ) are metric spaces and f : X → Y is a function. Then f is continuous at a ∈ X if and only if for all sequences a n → a in X, the sequence f(a n ) → f(a) in Y. 3. A metric space X is compact if and only if X is sequentially compact i.e. every sequence in X has a convergent subsequence. But in topological space only one part of 1. and 2. are true but the other part is not true, and both parts of 3. are not true. 1′. Suppose A is a nonem...