In
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, a dispersion point or explosion point is a point in a topological space the removal of which leaves the space highly disconnected.
More specifically, if ''X'' is a
connected topological space containing the
point
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Point ...
''p'' and at least two other points, ''p'' is a dispersion point for ''X'' if and only if
is
totally disconnected
In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...
(every subspace is disconnected, or, equivalently, every connected component is a single point). If ''X'' is connected and
is
totally separated
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties t ...
(for each two points ''x'' and ''y'' there exists a clopen set containing ''x'' and not containing ''y'') then ''p'' is an explosion point. A space can have at most one dispersion point or explosion point. Every totally separated space is totally disconnected, so every explosion point is a dispersion point.
The
Knaster–Kuratowski fan
In topology, a branch of mathematics, the Knaster–Kuratowski fan (named after Polish mathematicians Bronisław Knaster and Kazimierz Kuratowski) is a specific connected topological space with the property that the removal of a single point ...
has a dispersion point; any space with the
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 collecti ...
has an explosion point.
If ''p'' is an explosion point for a space ''X'', then the totally separated space
is said to be ''pulverized''.
References
*. (Note that this source uses ''hereditarily disconnected'' and ''totally disconnected'' for the concepts referred to here respectively as totally disconnected and totally separated.)
Topology
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