Holeyness

[…] the final rags soaked well, then washed by hand to scrub and scrub until the fabric died of holeyness.

Source: wiktionary

But no topologist, however accomplished, can separate a doughnut from its hole. So holeyness is a topological property.

Source: wiktionary

After my dishcloths attain holeyness, I give them a decent burial in my compost pile.

Source: wiktionary