Alertness is, not identical with being awake, since being awake is only one dimension of Alertness. It is the outermost dimension of Alertness, its surface only.
Three dimensions of Alertness may be identified. The surface, that is, the outermost dimension of Alertness is when the focus of attention is open the widest. Being Alert then means that now, in this very moment, with our eyes closed (or open) you pay attention to the processes of your inner world (bodily sensations, the stream of your thoughts, the shifting of your emotions), and the external world surrounding you (noises, scents etc. from the direct world around you). In such an instant you only focus your attention on what takes place in that very moment.