I got two definitions of $\displaystyle \sigma$-compact: one is from Munkres' "Topology" page 289, ex. 10, as the image below;

the other is from wikipedia, which says $\displaystyle \sigma$-compact means the space is the union of countably many compact subspaces.

I can deduce from Munkres' version to wikipedia's version, but I can not do the converse. Can the converse hold? If not, which one is the right definition? Thanks.