Geodesic
On Ditkin sets
Colloquium Mathematicum, Tome 69 (1996) no. 2, pp. 271-274.

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In the study of spectral synthesis S-sets and C-sets (see Rudin [3]; Reiter [2] uses the terminology Wiener sets and Wiener-Ditkin sets respectively) have been discussed extensively. A new concept of Ditkin sets was introduced and studied by Stegeman in [4] so that, in Reiter's terminology, Wiener-Ditkin sets are precisely sets which are both Wiener sets and Ditkin sets. The importance of such sets in spectral synthesis and their connection to the C-set-S-set problem (see Rudin [3]) are mentioned there. In this paper we study local properties, unions and intersections of Ditkin sets. (Warning: Usually in the literature "Ditkin set" means "C-set", but we follow the terminology of Stegeman.) Our results include: (i) if each point of a closed set E has a closed relative Ditkin neighbourhood, then E is a Ditkin set; (ii) any closed countable union of Ditkin sets is a Ditkin set; (iii) if $E_1 ∩ E_2$ is a Ditkin set, then $E_1 ∩ E_2$ is a Ditkin set if and only if $E_1$ and $E_2$ are Ditkin sets; and (iv) if $E_1, E_2$ are Ditkin sets with disjoint boundaries then $E_1 ∩ E_2$ is a Ditkin set.
DOI : 10.4064/cm-69-2-271-274

T. Muraleedharan 1 ; K. Parthasarathy 1

1 
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T. Muraleedharan; K. Parthasarathy. On Ditkin sets. Colloquium Mathematicum, Tome 69 (1996) no. 2, pp. 271-274. doi : 10.4064/cm-69-2-271-274. http://geodesic.mathdoc.fr/articles/10.4064/cm-69-2-271-274/

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