English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

A space $X$ is called a $k_{R}$-space, if $X$ is Tychonoff and the necessaryand sufficient condition for a real-valued function $f$ on $X$ to be continuousis that the restriction of $f$ to each compact subset is continuous. In thispaper, we discuss the $k_{R}$-property of products of sequential fans and freeAbelian topological groups by applying the $\kappa$-fan introduced by Banakh.In particular, we prove the following two results:  (1) The space $S_{\omega_{1}}\times S_{\omega_{1}}$ is not a $k_{R}$-space.  (2) The space $S_{\omega}\times S_{\omega_{1}}$ is a $k_{R}$-space if andonly if $S_{\omega}\times S_{\omega_{1}}$ is a $k$-space if and only if$\mathfrak b>\omega_1$.  These results generalize some well-known results on sequential fans.Furthermore, we generalize some results of Yamada on the free Abeliantopological groups by applying the above results. Finally, we pose some openquestions about the $k_{R}$-spaces.

The $k_{R}$-property of free Abelian topological groups and products of sequential fans