TY - JOUR
AU - Acharyya, Sudip Kumar
AU - Chattopadhyay, Kshitish Chandra
AU - Rooj, Pritam
PY - 2015/02/10
Y2 - 2024/09/17
TI - A generalized version of the rings CK(X) and Câˆž(X)– an enquery about when they become Noetheri
JF - Applied General Topology
JA - Appl. Gen. Topol.
VL - 16
IS - 1
SE -
DO - 10.4995/agt.2015.3247
UR - https://polipapers.upv.es/index.php/AGT/article/view/3247
SP - 81-87
AB - <p>Suppose F is a totally ordered field equipped with its order topology and X a completely F-regular topological space. Suppose P is an ideal of closed sets in X and X is locally-P. Let C<sub>P</sub>(X,F) ={f:Xâ†’F|f is continuous on X and its support belongs to P} and C<sup>P</sup><sub>âˆž</sub>(X,F) ={fâˆˆC<sub>P</sub>(X,F)| âˆ€ε>0 in F, cl<sub>X</sub>{xâˆˆX:|f(x)|> ε} âˆˆ P}. Then C<sub>P</sub>(X,F) is a Noetherian ring if and only if C<sup>P</sup><sub>âˆž</sub> (X,F) is a Noetherian ring if and only if X is a finite set. The fact that a locally compact Hausdorff space X is finite if and only if the ring C<sub>K</sub>(X) is Noetherian if and only if the ring C<sub>âˆž</sub>(X) is Noetherian, follows as a particular case on choosing F=R and P= the ideal of all compact sets in X. On the other hand if one takes F=R and P= the ideal of closed relatively pseudocompact subsets of X, then it follows that a locally pseudocompact space X is finite if and only if the ring C<sub>ψ</sub>(X) of all real valued continuous functions on X with pseudocompact support is Noetherian if and only if the ring C<sup>ψ</sup><sub>âˆž</sub>(X) ={fâˆˆC(X)| âˆ€ε >0, clX{xâˆˆX:|f(x)|> ε} is pseudocompact } is Noetherian. Finally on choosing F=R and P= the ideal of all closed sets in X, it follows that: X is finite if and only if the ring C(X) is Noetherian if and only if the ring C<sup>âˆ—</sup>(X) is Noetherian.</p>
ER -