In topology

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 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

Connected space

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 nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

 containing the point

Point (geometry)

In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

 p and at least two other points, p is a dispersion point for X if and only if is totally disconnected (every subspace is disconnected, or, equivalently, every connected component is a single point). If X is connected and is totally separated (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 has a dispersion point; any space with 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...

has an explosion point.

If p is an explosion point for a space X, then the totally separated space is said to be pulverized.