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We get the following result. A topological space is strongly paracompact if and only if for any monotone increasing open cover of it there exists a star-finite open refinement. We positively answer a question of the strongly paracompact property.

About Lehmer’s number, many people have studied its various properties, and obtained a series of interesting results. In this paper, we consider a generalized Lehmer problem: Let $p$ be a prime, and let $N(k;p)$ denote the number of all $1\le a_i\le p-1$ such that $a_1{a}_2\cdots a_k\equiv 1modp$ and $2\mid a_i+{\overline{a}}_i+1,$ $i=1,2,\cdots ,k$. The main purpose of this paper is using the analytic method, the estimate for character sums and trigonometric sums to study the asymptotic properties of the counting function $N(k;p),$ and give an interesting asymptotic formula for it.