In the following text for a discrete finite nonempty set and a self-map we investigate interaction between different entropies of generalized shift , and cellularities of some Alexandroff topologies on .
Let be the set of zero divisor elements of a commutative ring with identity and be the space of minimal prime ideals of with Zariski topology. An ideal of is called strongly dense ideal or briefly -ideal if and is contained in no minimal prime ideal. We denote by , the set of all for which is compact. We show that has property and is compact if and only if has no -ideal. It is proved that is an essential ideal (resp., -ideal) if and only if is an almost locally compact...
A space is called -compact by M. Mandelker if the intersection of all free maximal ideals of coincides with the ring of all functions in with compact support. In this paper we introduce -compact and -compact spaces and we show that a space is -compact if and only if it is both -compact and -compact. We also establish that every space admits a -compactification and a -compactification. Examples and counterexamples are given.
Suppose that K is a CW-complex, X is an inverse sequence of stratifiable spaces, and X = limX. Using the concept of semi-sequence, we provide a necessary and sufficient condition for X to be an absolute co-extensor for K in terms of the inverse sequence X and without recourse to any specific properties of its limit. To say that X is an absolute co-extensor for K is the same as saying that K is an absolute extensor for X, i.e., that each map f:A → K from a closed subset A of X extends to a map F:X...
In [12] Petrunin proves that a compact metric space X admits an intrinsic isometry into En if and only if X is a pro-Euclidean space of rank at most n, meaning that X can be written as a “nice” inverse limit of polyhedra. He also shows that either case implies that X has covering dimension at most n. The purpose of this paper is to extend these results to include both embeddings and spaces which are proper instead of compact. The main result of this paper is that any pro-Euclidean space of rank...