*Definition [#ced989c5]
Suppose that if U is an open cover of a topological space X, there exests a sequence &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b$V_n$%7d%25.png); of open covers and for every point x in X, there exists an index m and W in U such that &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b$%5cmathrm%7bSt%7d%28x%2cV_m%29%5csubset%20W$%7d%25.png);.
Then X is called σ-compact.

*Property [#n2761eb1]
-Let X be a topological space. The following are equivalent.
++X is σ-paracompact.
++Every open cover of X has a [[σ-discrete]] closed refinement.
++Every open cover of X has a [[σ-locally-finite]] closed refinement.
++Every open cover of X has a [[σ-closure-preserving]] closed refinement.
*Reference [#mcd46ab1]
Dennis K. Burke, ''On subcompact spaces'', Proc. Amer. Math. Soc. Vol.23 No.3 (1969) pp.655-663.