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High Quality Content by WIKIPEDIA articles! In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology (or the relative topology, or the induced topology, or the trace topology). The subspace topology of the natural numbers, as a subspace of R, is the discrete topology. The rational numbers Q considered as a subspace of R do not have the discrete topology (the point 0 for example is not an open set in Q). If a and b are rational, then the intervals (a, b) and [a, b] are respectively open and closed, but if a and b are irrational, then the set of all x with a x b is both open and closed. The set [0,1] as a subspace of R is both open and closed, whereas as a subset of R it is only closed.