Geodesic
Pseudouniform topologies on $C(X)$ given by ideals
Commentationes Mathematicae Universitatis Carolinae, Tome 54 (2013) no. 4, pp. 557-577.

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Given a Tychonoff space $X$, a base $\alpha$ for an ideal on $X$ is called pseudouniform if any sequence of real-valued continuous functions which converges in the topology of uniform convergence on $\alpha$ converges uniformly to the same limit. This paper focuses on pseudouniform bases for ideals with particular emphasis on the ideal of compact subsets and the ideal of all countable subsets of the ground space.
Classification : 54A10, 54A20, 54A25, 54C35, 54D20, 54E15
Keywords: function space; topology of uniform convergence; ideal; uniformity; Lindelöf property; pseudouniform ideal; almost pseudo-$\omega$-bounded 
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Pichardo-Mendoza, Roberto; Tamariz-Mascarúa, Ángel; Villegas-Rodríguez, Humberto. Pseudouniform topologies on $C(X)$ given by ideals. Commentationes Mathematicae Universitatis Carolinae, Tome 54 (2013) no. 4, pp. 557-577. http://geodesic.mathdoc.fr/item/CMUC_2013__54_4_a6/