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| In [[topology]], a '''dispersion point''' or '''explosion point''' is a point in a topological space the removal of which leaves the space highly disconnected.
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| More specifically, if ''X'' is a [[connected space|connected topological space]] containing the [[point (geometry)|point]] ''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]] (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]] (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.
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| The [[Knaster–Kuratowski fan]] has a dispersion point; any space with the [[particular point topology]] has an explosion point.
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| 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''.
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| ==References==
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| *{{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=http://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.)
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| [[Category:Topology]]
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| {{topology-stub}}
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