Continuum (topology)

In the mathematical field of point-set topology, a continuum (plural: "continua") is a nonempty compact connected metric space, or, less frequently, a compact connected Hausdorff space. Continuum theory is the branch of topology devoted to the study of continua.



Warsaw circle


There are two fundamental techniques for constructing continua, by means of nested intersections and inverse limits.

  • If {Xn} is a nested family of continua, i.e. Xn Xn+1, then their intersection is a continuum.
  • If {(Xn, fn)} is an inverse sequence of continua Xn, called the coordinate spaces, together with continuous maps fn: Xn+1 Xn, called the bonding maps, then its inverse limit is a continuum.

A finite or countable product of continua is a continuum.

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