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.
Extension topology.
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".
Open extension topology.
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 "X" ∪ "Q", where "Q" is a subset of "P", or "A", where "A" is an open set of "X".
For this reason this topology is called the open extension topology of "X" plus "P", with which one extends to "X" ∪ "P" the open 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 closed sets in "X" ∪ "P" are of the form: "Q", where "Q" is a subset of "P", or "B" ∪ "P", where "B" is a closed set of "X". Note that "P" is closed in "X" ∪ "P" and "X" is open and dense in "X" ∪ "P".
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 "X" ∪ "P" is smaller than the extension topology of "X" ∪ "P".
Assuming "X" and "P" are not empty to avoid trivialities, here are a few general properties of the open extension topology:
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".