Dispersion point

Template:Short description In topology, a dispersion pointTemplate:SfnTemplate:Sfn 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,Template:Sfn
• 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 Sierpiński space and an 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 pointTemplate:Sfn 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.Template:Sfn

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

• Template:Citation. (Note that this source uses hereditarily disconnected and totally disconnected for the concepts referred to here respectively as totally disconnected and totally separated.)
• Template:Cite book