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 .

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Concepts associés (2)

List of topologies

The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property. Discrete topology − All subsets are open. Indiscrete topology, chaotic topology, or Trivial topology − Only the empty set and its complement are open.

Particular point topology

In mathematics, the particular point topology (or included point topology) is a topology where a set is open if it contains a particular point of the topological space. Formally, let X be any non-empty set and p ∈ X. The collection of subsets of X is the particular point topology on X. There are a variety of cases that are individually named: If X has two points, the particular point topology on X is the Sierpiński space. If X is finite (with at least 3 points), the topology on X is called the finite particular point topology.