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If M is a set of points, and P is a point of M, then the quasi-component of M determined by P is the set of points common to all possible sets Mi, where M = M1 + M2, Mi contains P, and Mi and Mi are mutually separated sets.f If M is a connected set, then the quasi-component of M determined by P is M itself. If M is not connected, then it can be considered as the sum of its quasi-components. If a quasi-component consists of more than one point, it is called a true quasi-component. If all the quasi-components of M reduce to single points, that is, if M contains no true quasi-components, then M is totally disconnected (i. e., has no connected subset consisting of more than one point). However, the quasi-components of a totally disconnected set do not necessarily reduce to single points. Sierpinski has givenj an example of a set of points N and a point P not in N, such that N has no true quasi-components, and such that the set N+P, although totally disconnected, contains a true quasi-component consisting of P and a certain point of N. Knaster and Kuratowski have given § an example of a totally disconnected set of points S, which becomes connected upon the addition of a certain point a; i. e., S+a

A point set which has no true quasicomponents, and which becomes connected upon the addition of a single point