Dispersion point

Short description: Point in a topological space

In topology, a dispersion point of a topological space $X$ is a point $p\in X$ such that $X$ is connected and removing the point leaves $X\setminus \{p\}$ totally disconnected (meaning that its connected components are all singletons, i.e., $X\setminus \{p\}$ contains no connected subset of size at least two).

Examples of spaces with a dispersion point include:

• the Knaster–Kuratowski fan,
• spaces with a particular point topology,
• spaces with an excluded point topology.

A connected space $X$ with $|X|\ge 3$ can have at most one dispersion point. The case $|X|=2$ (covering the Sierpiński space and the indiscrete space of size two) is a degenerate case, where every point of $X$ is a dispersion point (because removing a point leaves a singleton, which is totally disconnected).

Explosion point

An explosion point of a topological space $X$ is a point $p\in X$ such that $X$ is connected and removing the point leaves $X\setminus \{p\}$ totally separated (meaning that for any two distinct points $x$ and $y$ there is a clopen set containing $x$ and not $y$).

Every explosion point is a dispersion point (because totally separated spaces are totally disconnected).

Examples of spaces with an explosion point include spaces with a particular point topology and spaces with an excluded point topology. The Knaster–Kuratowski fan has a dispersion point which is not an explosion point.

A space of size at least three can have at most one explosion point.

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

References

• Abry, Mohammad; Dijkstra, Jan J.; van Mill, Jan (2007), "On one-point connectifications", Topology and Its Applications 154 (3): 725–733, doi:10.1016/j.topol.2006.09.004, https://www.math.vu.nl/~vanmill/papers/papers2007/abry.pdf . (Note that this source uses hereditarily disconnected and totally disconnected for the concepts referred to here respectively as totally disconnected and totally separated.)
• Steen, Lynn Arthur; Seebach, J. Arthur (1978). Counterexamples in topology. New York, NY: Springer New York. doi:10.1007/978-1-4612-6290-9. ISBN 978-0-387-90312-5.