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 $$(X, \mathcal{T})$$ be a topological space and $$P$$ a set disjoint from $$X$$. The open extension topology of $$\mathcal{T}$$ plus $$P$$ is $$\mathcal{T}^* = \mathcal{T} \cup \{X \cup A : A \subset P\}.$$Let $$X^* = X \cup P$$. Then $$\mathcal{T}^*$$is a topology in $$X^*$$. The subspace topology of $$X$$ is the original topology of $$X$$, i.e. $$\mathcal{T}^*|X = \mathcal{T}$$, while the subspace topology of $$P$$ is the discrete topology, i.e. $$\mathcal{T}^*|P = \mathcal{P}(P)$$.

The closed sets in $$X^*$$ are $$\{B \cup P : X \subset B \land X \setminus B \in \mathcal{T}\}$$. Note that $$P$$ is closed in $$X^*$$ and $$X$$ is open and dense in $$X^*$$.

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 $$X^*$$ is smaller than the extension topology of $$X^*$$.

Assuming $$X$$ and $$P$$ are not empty to avoid trivialities, here are a few general properties of the open extension topology:
 * $$X$$ is dense in $$X^*$$.
 * If $$P$$ is finite, $$X^*$$ is compact. So $$X^*$$ is a compactification of $$X$$ in that case.
 * $$X^*$$ is connected.
 * If $$P$$ has a single point, $$X^*$$ 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.