# Dispersion point

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{{Short description|Point in a topological space}}
In [topology](/source/topology), a '''dispersion point''' or '''explosion point''' is a point in a topological space the removal of which leaves the space totally disconnected.

More specifically, if ''X'' is a [connected topological space](/source/connected_space) containing the [point](/source/point_(geometry)) ''p'' and at least two other points, ''p'' is a dispersion point for ''X'' if and only if <math>X\setminus \{p\}</math> is [totally disconnected](/source/totally_disconnected) (every subspace is disconnected, or, equivalently, every connected component is a single point). If ''X'' is connected and <math>X\setminus \{p\}</math> is [totally separated](/source/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](/source/Knaster%E2%80%93Kuratowski_fan) has a dispersion point; any space with the [particular point topology](/source/particular_point_topology) has an explosion point.

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

==References==
*{{citation|first1=Mohammad|last1=Abry|first2=Jan J.|last2=Dijkstra|first3=Jan|last3=van Mill|title=On one-point connectifications|journal=Topology and Its Applications|volume=154|issue=3|year=2007|pages=725–733|doi=10.1016/j.topol.2006.09.004|url=https://www.math.vu.nl/~vanmill/papers/papers2007/abry.pdf|doi-access=free}}. (Note that this source uses ''hereditarily disconnected'' and ''totally disconnected'' for the concepts referred to here respectively as totally disconnected and totally separated.)

Category:Topology

{{topology-stub}}

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Adapted from the Wikipedia article [Dispersion point](https://en.wikipedia.org/wiki/Dispersion_point) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Dispersion_point?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.