Subspace topology

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 topology induced from that of X called the subspace topology (or the relative topology, or the induced topology, or the trace topology).

Given a topological space ${\displaystyle (X,\tau )}$ and a subset ${\displaystyle S}$ of ${\displaystyle X}$, the subspace topology on ${\displaystyle S}$ is defined by

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