Close shaven, saving face (7)

I believe the answer is:

airless

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'close' is the definition.
(I know that close can be written as airless)

I can't explain the rest of the clue.

(Other definitions for airless that I've seen before include "Very stuffy" , "not a breath of wind" , "Close" , "Unventilated" , "Lacking ventilation" .)

I've seen this clue in The Times.

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