Continuum

2020 Mathematics Subject Classification: Primary: 54A05,54E45 [MSN][ZBL]

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).

Every locally connected metric continuum is a continuous image of a closed segment (the Hahn–Mazurkiewicz theorem). A non-degenerate continuum is irreducible between two of its points if no proper subcontinuum contains these two points. A continuum that is irreducible between any two of its points is called an irreducible continuum. Every locally connected irreducible continuum is a simple arc, that is, is homeomorphic to an interval.

An irreducible continuum is said to be indecomposable if it cannot be represented as the sum of two proper subcontinua, and is called hereditarily indecomposable if the continuum itself and all of its subcontinua are indecomposable. A continuum is indecomposable if and only if it contains three points such that it is irreducible between any two of them; all hereditarily-indecomposable metrizable chainable continua are homeomorphic.

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