Y – R plus R is the same as the original topology of Y, and the answer is in general no. Note that the open extension topology of is smaller than the extension topology of . Assuming and are not empty to avoid trivialities, here are a few general properties of the open extension topology: is dense in . If is finite, is compact. So is a compactification of in that case. is connected. If has a single point, is ultraconnected. For a set Z and a point p in Z, one obtains the excluded point topology construction by considering in Z the discrete topology and applying the open extension topology construction to Z – {p} plus p. Closed extension topology Let X be a topological space and P a set disjoint from X. Consider in X ∪ P the topology whose closed sets are of the form X ∪ Q, where Q is a subset of P, or B, where B is a closed set of X. For this reason this topology is called the closed extension topology of X plus P, with which one extends to X ∪ P the closed sets of X. As subsets of X ∪ P the subspace topology of X is the original topology of X, while the subspace topology of P is the discrete topology. The open sets of X ∪ P are of the form Q, where Q is a subset of P, or A ∪ P, where A is an open set of X. Note that P is open in X ∪ P and X is closed in X ∪ P. If Y is a topological space and R is a subset of Y, one might ask whether the closed extension topology of Y – R plus R is the same as the original topology of Y, and the answer is in general no. Note that the closed extension topology of X ∪ P is smaller than the extension topology of X ∪ P. For a set Z and a point p in Z, one obtains the particular point topology construction by considering in Z the discrete topology and applying the closed extension topology construction to Z – {p} plus p. Notes Works cited Topological spaces Topology