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Closure theorem for partially semialgebraic sets

Łojasiewicz S., Zurro M. A.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

Vol. 55, no 4

325-331

In 1988 it was proved by the first author that the closure of a partially semialgebraic set is partially semialgebraic. The essential tool used in that proof was the regular separation property. Here we give another proof without using this tool, based on the semianalytic L-cone theorem (Theorem 2), a semianalytic analog of the Cartan-Remmert-Stein lemma with parameters.

On the gradient inequality

Łojasiewicz S., Zurro M.

Bulletin of the Polish Academy of Sciences. Mathematics

1999

Vol. 47, no 2

143-145

Some proof on the Gradient Inequality is given.