Closed extension topology
Encyclopedia
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.
Note that 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. Note that the subspace topology
of X as a subset of X ∪ P is the original topology of X, while the subspace topology of P as a subset of X ∪ P is the discrete topology
.
Being Y a topological space and R 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 similitude 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.
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. Note that the subspace topology
of X as a subset of X ∪ P is the original topology of X, while the subspace topology of P as a subset of X ∪ P is the discrete topology
.
Note that the closed sets of 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.
Being Y a topological space and R 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 that the open extension topology of X ∪ P is smaller
than the extension topology of X ∪ P.
Being Z a set and p a point 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.
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. Note that the subspace topology
of X as a subset of X ∪ P is the original topology of X, while the subspace topology of P as a subset of X ∪ P is the discrete topology
.
Note that 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.
Being Y a topological space and R 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 that the closed extension topology of X ∪ P is smaller
than the extension topology of X ∪ P.
Being Z a set and p a point 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.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an extension topology is a topology placed on the disjoint union
Disjoint union
In mathematics, the term disjoint union may refer to one of two different concepts:* In set theory, a disjoint union is a modified union operation that indexes the elements according to which set they originated in; disjoint sets have no element in common.* In probability theory , a disjoint union...
of a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
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.Note that 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. Note that the subspace topology
Subspace topology
In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology .- Definition :Given a topological space and a subset S of X, the...
of X as a subset of X ∪ P is the original topology of X, while the subspace topology of P as a subset of X ∪ P is the discrete topology
Discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...
.
Being Y a topological space and R 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 similitude 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. Note that the subspace topology
Subspace topology
In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology .- Definition :Given a topological space and a subset S of X, the...
of X as a subset of X ∪ P is the original topology of X, while the subspace topology of P as a subset of X ∪ P is the discrete topology
Discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...
.
Note that the closed sets of 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.
Being Y a topological space and R 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 that the open extension topology of X ∪ P is smaller
Comparison of topologies
In topology and related areas of mathematics comparison of topologies refers to the fact that two topological structures on a given set may stand in relation to each other. The set of all possible topologies on a given set forms a partially ordered set...
than the extension topology of X ∪ P.
Being Z a set and p a point in Z, one obtains the excluded point topology
Excluded point topology
In mathematics, the excluded point topology is a topology where exclusion of a particular point defines openness. Formally, let X be any set and p ∈ X. The collectionof subsets of X is then the excluded point topology on X....
construction by considering in Z the discrete topology
Discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...
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. Note that the subspace topology
Subspace topology
In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology .- Definition :Given a topological space and a subset S of X, the...
of X as a subset of X ∪ P is the original topology of X, while the subspace topology of P as a subset of X ∪ P is the discrete topology
Discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...
.
Note that 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.
Being Y a topological space and R 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 that the closed extension topology of X ∪ P is smaller
Comparison of topologies
In topology and related areas of mathematics comparison of topologies refers to the fact that two topological structures on a given set may stand in relation to each other. The set of all possible topologies on a given set forms a partially ordered set...
than the extension topology of X ∪ P.
Being Z a set and p a point in Z, one obtains the particular point topology
Particular point topology
In mathematics, the particular point topology is a topology where sets are considered open if they are empty or contain a particular, arbitrarily chosen, point of the topological space. Formally, let X be any set and p ∈ X. The collectionof subsets of X is then the particular point topology...
construction by considering in Z the discrete topology
Discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...
and applying the closed extension topology construction to Z - {p} plus p.