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Monotone determined spaces are natural topological extensions of dcpo. Itsmain purpose is to build an extended framework for domain theory. In thispaper, we study the one-step closure and ideal convergence on monotonedetermined space. Then we also introduce the equivalent characterizations ofc-spaces and locally hypercompact space. The main results are:1.Every c-spacehas one-step closure and every locally hypercompact space has weak one-stepclosure;2.A monotone determined space has one-step closure if and only if it isd-meet continuous and has weak one-step closure. 3.IS-convergence(resp.IGS-convergence) is topological iff X is a c-space (resp. locally hypercompactspace); 4.If X is a d-meet continuous space, then the following threeconditions are equivalent to each other: (i) X is c-space; (ii) The net (xj )ISL-converges to x iff (xj ) I-converges to x with respect to Lawson topology;(iii) The net (xj ) IGSL-converges to x iff (xj ) I-converges to x with respectto Lawson topology.

One-step Closure, Ideal Convergence and Monotone Determined Space