Extension topology explained

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

The closed sets in

X ^{*} are \{B \cup P : X \subset B \land X \setminus B \in l{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 – 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 – plus p.