Extension topology

In topology, a branch of mathematics, an extension topology is a topology placed on the disjoint union of a topological space and another set. There are various types of extension topology, described in the sections below. Let X be a topological space and P a set disjoint from X. Consider in X ∪ P the topology whose open sets are of the form A ∪ Q, where A is an open set of X and Q is a subset of P. The closed sets of X ∪ P are of the form B ∪ Q, where B is a closed set of X and Q is a subset of P. For these reasons this topology is called the extension topology of X plus P, with which one extends to X ∪ P the open and 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. As a topological space, X ∪ P is homeomorphic to the topological sum of X and P, and X is a clopen subset of X ∪ P. If Y is a topological space and R is a subset of Y, one might ask whether the extension topology of Y – R plus R is the same as the original topology of Y, and the answer is in general no. Note the similarity of this extension topology construction and the Alexandroff one-point compactification, in which case, having a topological space X which one wishes to compactify by adding a point ∞ in infinity, one considers the closed sets of X ∪ {∞} to be the sets of the form K, where K is a closed compact set of X, or B ∪ {∞}, where B is a closed set of X. Let be a topological space and a set disjoint from . The open extension topology of plus is Let . Then is a topology in . The subspace topology of is the original topology of , i.e. , while the subspace topology of is the discrete topology, i.e. . The closed sets in are . Note that is closed in and is open and dense in . If Y a topological space and R is a subset of Y, one might ask whether the open 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 open extension topology of is smaller than the extension topology of .