# Continuum

A non-empty connected compact Hausdorff space (cf. Compact space). A continuum is said to be degenerate if it consists of a single point. Of special importance is the class of metrizable continua. Examples of continua: a closed segment, a circle, a convex polytope, etc. A Hausdorff compactum $(X,\rho)$ (that is, a metrizable compactum with metric $\rho$) is a continuum if and only if for every pair of points $a, b \in X$ and for any $\epsilon > 0$ there is a finite $\epsilon$-chain joining these points, that is, a sequence $\{x_n\}_{n=1}^k$ of points in $X$ such that $x_1 = a$, $x_k = b$ and $\rho(x_n,x_{n+1} )< \epsilon$. The union of two continua having a point in common is a continuum. The topological product of continua is a continuum, a continuous image of a continuum is a continuum, the components of a Hausdorff compactum are continua, the intersection of a decreasing sequence of continua is a continuum. No continuum can be decomposed into a countable union of non-empty disjoint closed sets (Sierpiński's theorem).