That is the daughters' excuse for getting dear old dad drunk, then raping him in his sleep; but Zoar, 'the small city', was spared for Lot's sake, as a refuge for him...then (AFTER the destruction of the OTHER cities) he went into the very hills he had earlier told the angels that he would not be able survive in.
Gen 19:19-24
19 Behold now, thy servant hath found grace in thy sight, and thou hast magnified thy mercy, which thou hast shewed unto me in saving my life; and I cannot escape to the mountain, lest some evil take me, and I die:
20 Behold now, this city is near to flee unto, and it is a little one: Oh, let me escape thither, (is it not a little one?) and my soul shall live.
21 And he said unto him, See, I have accepted thee concerning this thing also, that I will not overthrow this city, for the which thou hast spoken.
22 Haste thee, escape thither; for I cannot do any thing till thou be come thither. Therefore the name of the city was called Zoar.
23 The sun was risen upon the earth when Lot entered into Zoar.
24 Then the Lord rained upon Sodom and upon Gomorrah brimstone and fire from the Lord out of heaven;
Gen 19:30-31
30 And Lot went up out of Zoar, and dwelt in the mountain, and his two daughters with him; for he feared to dwell in Zoar: and he dwelt in a cave, he and his two daughters.
31 And the firstborn said unto the younger, Our father is old, and there is not a man in the earth to come in unto us after the manner of all the earth:
Though it doesn't explain WHY he 'feared to live in Zoar, it may be that the people blamed HIM, rather than their sins, for the destruction of the other cities. That could also be why the daughters knew that there was 'no man to come unto them': they were persona non grata; not there there wasn't any man alive in the area.
It could have been that they had the only cave with a porch in front with a couple of packs of dogs living under it.
I'm just saying...