When is the {I}sbell topology a group topology?

Conditions on a topological space $X$ under which the space $C(X,\mathbb{R})$ of continuous real-valued maps with the Isbell topology $\kappa$ is a topological group (topological vector space) are investigated. It is proved that the addition is jointly continuous at the zero function in $C_{\kappa}(X,\mathbb{R})$ if and only if $X$ is infraconsonant. This property is (formally) weaker than consonance, which implies that the Isbell and the compact-open topologies coincide. It is shown the translations are continuous in $C_{\kappa}(X,\mathbb{R})$ if and only if the Isbell topology coincides with the fine Isbell topology. It is proved that these topologies coincide if $X$ is prime (that is, with at most one non-isolated point), but do not even for some sums of two consonant prime spaces

Function spaces and contractive extensions in Approach Theory: The role of regularity

Two classical results characterizing regularity of a convergence space in terms of continuous extensions of maps on one hand, and in terms of continuity of limits for the continuous convergence on the other, are extended to convergence-approach spaces. Characterizations are obtained for two alternative extensions of regularity to convergence-approach spaces: regularity and strong regularity. The results improve upon what is known even in the convergence case. On the way, a new notion of strictness for convergence-approach spaces is introduced.Comment: previous version had an error, fixed here with a new definition of strictnes

Strongly sequential spaces

summary:The problem of Y. Tanaka [10] of characterizing the topologies whose products with each first-countable space are sequential, is solved. The spaces that answer the problem are called strongly sequential spaces in analogy to strongly Fréchet spaces

A Discussion of a Flipped classroom Model (for Calculus), and of the Use of Self-Graded Homework

The intent is to share my experience with a flipped Calculus course using resources I have developed: instructional videos, e-books, and a comprehensive set of resources for self-graded homework. I have also used this technique successfully for other non-flipped courses, and this may be of interest whether one wants to flip the classroom or not