By Ghora J.K.

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**Additional resources for A Central Limit Theorem for the L2 Error of Positive Wavelet Density Estimator**

**Example text**

Ii) A closure operator is a projection c on L with 1L

3) implies (4): By (3), d is a coretraction, hence, it is injective. 6 we have d Is = gd. ~ dgd, and if d is injective, we have Thus, g is a retraction and hence surjective. The equivalence of (1*)-(4*) is proved dually. 0 We indicated in earlier examples how closure and kernel operators function in applications. 8. DEFINmON. Let L be a poset. A projection is an idempotent, monotone self map p : L-+ L. (ii) A closure operator is a projection c on L with 1L

7(4). A. A is complete. A. We note, too, that F«F holds iff F is finitely generated. This and related examples will be studied in full detail in Section 4 on algebraic lattices. 0 Topological spaces provide other good examples-in certain cases. That is to say, in certain cases it is easy to identify the way-below relation in topological tenns. In our fonnulations we adopt the Bourbaki convention of calling a space (or subset) with the Heine-Borel property quasicompact and reserve the adjective compact for the Hausdorff setting.