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A topological space  X  has property      σ  −  CM \sigma - {\text {CM}}       if for every countable family  F  of continuous scalar valued functions on  X  there is a compact metrizable subspace  M  of  X  such that      f ( X ) = f ( M ) f(X) = f(M)       for every  f  in  F . Every compact metric space, every weakly compact subset of a Banach space and every closed ordinal space has property      σ  −  CM \sigma - {\text {CM}}      . Every continuous image of an arbitrary product of spaces having property      σ  −  CM \sigma - {\text {CM}}       also has property      σ  −  CM \sigma - {\text {CM}}      . If  X  has property      σ  −  CM \sigma - {\text {CM}}      , then every copy of      c 0 {c_0}       in      C ( X ) C(X)       is complemented in      C ( X ) C(X)      . If a locally convex space  E  belongs to the variety of locally convex spaces generated by the weakly compactly generated Banach spaces, then every copy of      c 0 {c_0}       in  E  is complemented in  E .

On the existence of compact metric subspaces with applications to the complementation of 𝑐₀