keep an eye on vs check out

keep an eye on and check out both are verbs.

Word	Noun	Adjective	Verb	Adverb
keep an eye on	No	No	Yes	No
check out	No	No	Yes	No

As verbs, check out is a hypernym of keep an eye on; that is, check out is a word with a broader meaning than keep an eye on:

Other hypernyms of keep an eye on include check, check into, check over, check up on, go over, look into, suss out.

keep an eye on (verb)	check out (verb)
follow with the eyes or the mind	examine so as to determine accuracy, quality, or condition
	announce one's departure from a hotel
	try to learn someone's opinions and intentions
	record, add up, and receive payment for items purchased
	withdraw money by writing a check
	trace
	be verified or confirmed; pass inspection