A note regarding “contractibility”, “contractibility with respect to” and “contractibility in”

In this article, we present a comparative study of the concepts of contractibility, contractibility with respect to, and contractibility in, applied in topological spaces and in hyperspaces of continua. The aim is to clarify and correct certain statements that appear in some works where these terms are used, either because they contain inaccuracies or inconsistencies, or because they are employed in a manner different from their original introduction. Through this, we seek to clarify the distinctions between their definitions and to facilitate proper understanding for the reader.