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 ${\displaystyle X\setminus \{p\}}$ is totally disconnected (every subspace is disconnected, or, equivalently, every connected component is a single point). If X is connected and ${\displaystyle X\setminus \{p\}}$ 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 ${\displaystyle X\setminus \{p\}}$ 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.)