# Dispersion point

In topology, 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 *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 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

- Abry, Mohammad; Dijkstra, Jan J.; van Mill, Jan (2007), "On one-point connectifications" (PDF),
*Topology and its Applications*,**154**(3): 725–733, doi:10.1016/j.topol.2006.09.004. (Note that this source uses*hereditarily disconnected*and*totally disconnected*for the concepts referred to here respectively as totally disconnected and totally separated.)