Let $\lbrace X_\alpha \rbrace _{\alpha \in \Lambda }$ be a family of topological spaces and $x_{\alpha }\in X_{\alpha }$, for every $\alpha \in \Lambda $. Suppose $X$ is the quotient space of the disjoint union of $X_\alpha $’s by identifying $x_\alpha $’s as one point $\sigma$. We try to characterize ideals of $C(X)$ according to the same ideals of $C(X_\alpha )$’s. In addition we generalize the concept of rank of a point, see [9], and then answer the following two algebraic questions. Let $m$ be an infinite cardinal. (1) Is there any ring $R$ and $I$ an ideal in $R$ such that $I$ is an irreducible intersection of $m$ prime ideals? (2) Is there any set of prime ideals of cardinality $m$ in a ring $R$ such that the intersection of these prime ideals can not be obtained as an intersection of fewer than $m$ prime ideals in $R$? Finally, we answer an open question in [11].
Chequered blue butterfly, Scolitantides orion (Lepidoptera: Lycaenidae) has severely declined in many parts of Europe and is currently red-listed in many countries. We studied the population structure and turnover of the species in a lake-island system in a National Park in eastern Finland over a three-year period. The incidence of the chequered blue on the suitable islands (n = 41) and habitat patches (n = 123) was high: an average of 82% of the islands and patches were occupied over the three year period. At the island scale, the annual population turnover rate was 17%, with an extinction and colonization rate of 7% and 10%, respectively. At the patch scale, the annual population turnover was 16%, with 7% extinction and 9% colonization rate. Islands that were occupied over the three year period had a larger area of suitable habitat than islands in which turnover events were observed. At the patch scale, turnover events were observed in small and poorly connected patches. Patchy occurrence of the host plant and observed extinction-colonization dynamics suggest that the chequered blue population confirms a metapopulation structure. Although the local populations are small, the observed high patch occupancy and balanced population turnover indicates that the metapopulation is not in immediate risk of extinction.