# Irreducible mapping

A continuous mapping of a topological space $X$ onto a topological space $Y$ such that the image of every closed set in $X$, other than $X$ itself, is different from $Y$. If $f : X \rightarrow Y$ is a continuous mapping, $f(X) = Y$, and if all inverse images of points under $f$ are compact, then there exists a closed subspace $X_1$ in $X$ such that $f(X_1) = Y$ and such that the restriction of $f$ to $X_1$ is an irreducible mapping. The combination of the requirements on a mapping of being irreducible and being closed has an outstanding effect: Spaces linked by such mappings do not differ in a number of important cardinal characteristics; in particular, they have the same Suslin number and $\pi$-weight. But the main value of closed irreducible mappings lies in the central role they play in the theory of absolutes.