A note on p-bounded and quasi-p-bounded subsets

Abstract

that bounded subsets of a GLOTS are strongly-bounded

We discuss the relationship between p-boundedness and quasi-p-boundedness in the realm of GLOTS for p is an element of omega. We show that p-pseudocompactness, p-compactness, quasi-p-pseudocompactness and quasi-p-compactness are equivalent properties for a GLOTS

and C-compact subsets of a GLOTS are strongly-C-compact. We also show that a topologically orderable group is locally precompact if and only if it is metrizable. For bounded subsets of a GLOTS, a version of the classical Gilcksberg's Theorem on pseudocompactness is obtained: if A(alpha) is a bounded subset of a GLOTS X-alpha for each alpha is an element of Delta, then cl(beta(Pialphais an element ofDelta) X-alpha) (Pi(alphais an element ofDelta) A(alpha)) = Pi(alphais an element ofDelta) cl(beta(Xalpha))A(alpha). Also we prove that there exists an ultrapseudocompact topological group which is not quasi-p-compact for any p is an element of omega. To see this example, p-pseudocompactness and p-compactness are investigated in the field of C-pi-spaces, proving that ultracompactness, quasi-p-compactness for a p is an element of omega and countable compactness (respectively, ultrapseudocompactness, quasi-p-pseudocompactness for a p is an element of omega and pseudocompactness) are equivalent properties in the class of spaces of the form C-pi(X, [0, 1]).