top of the inning vs bout

top of the inning and bout both are nouns.

Word	Noun	Adjective	Verb	Adverb
top of the inning	Yes	No	No	No
bout	Yes	No	No	No

As nouns, bout is a hypernym of top of the inning; that is, bout is a word with a broader meaning than top of the inning:

top of the inning: the first half of an inning; while the visiting team is at bat
bout: (sports) a division of a game during which one team is on the offensive

Other hypernyms of top of the inning include round, turn.

top of the inning (noun)	bout (noun)
the first half of an inning; while the visiting team is at bat	an occasion for excessive eating or drinking
	a contest or fight (especially between boxers or wrestlers)
	(sports) a division of a game during which one team is on the offensive
	a period of illness