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A net is a generalization of a sequence where a directed set is used as the index set instead of positive integers. Convergence of nets can be defined in a similar way as convergence of sequences. Convergent nets in a topological space uniquely determine its topology.
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Relative compactness in topological spaces (reference request)
My guess is that even $T_3$ is already sufficient. I do not have access to the monograph Fletcher, Peter and Lindgren, William F., Quasi-uniform spaces, M. Dekker, New York, Basel 1982, in the moment, …