Countable composition closedness and integer-valued continuous functions in pointfree topology

Banaschewski, Bernhard

Countable composition closedness and integer-valued continuous functions in pointfree topology

Document Type : Research Paper

Author

Bernhard Banaschewski

Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada.

Abstract

‎For any archimedean$f$-ring $A$ with unit in whichbreak$awedge‎ ‎(1-a)leq 0$ for all $ain A$‎, ‎the following are shown to be‎ ‎equivalent‎: ‎ ‎1‎. ‎$A$ is isomorphic to the $l$-ring ${mathfrak Z}L$ of all‎ ‎integer-valued continuous functions on some frame $L$‎. 2‎. ‎$A$ is a homomorphic image of the $l$-ring $C_{Bbb Z}(X)$‎ ‎of all integer-valued continuous functions‎, ‎in the usual sense‎, ‎on some topological space $X$‎. 3‎. ‎For any family $(a_n)_{nin omega}$ in $A$ there exists an‎ ‎$l$-ring homomorphism break$varphi‎ :‎C_{Bbb Z}(Bbb‎ ‎Z^omega)rightarrow A$ such that $varphi(p_n)=a_n$ for the‎ ‎product projections break$p_n:{Bbb Z^omega}rightarrow Bbb Z$‎. ‎This provides an integer-valued counterpart to a familiar result‎ ‎concerning real-valued continuous functions‎.

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References

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