[Sam Hill]

"The modern short story. Poe can be said to have invented the short story period."

Nah. Not even close. "Canterbury Tales" and "Decameron." Those were short story collections. There have been many many before Poe.

No, Poe invented the MODERN short story. He defined it in terms that Chekov also followed, wittingly or not.

Even Wikipedia agrees, and they know everything:

Short story - Wikipedia, the free encyclopedia

Modern short stories

Modern short stories emerged as their own genre in the early 19th century. Early examples of short story collections include the Brothers Grimm Fairy Tales (1824-1826), Nathaniel Hawthorne's Twice Told Tales (1842), Edgar Allan Poe's Tales of the Grotesque and Arabesque (1836), and Guy de Maupassant's La Maison Tellier (1881). In the later part of the 19th century, the growth of print magazines and journals created a strong market demand for short fiction between 3,000 and 15,000 words in length. Among the famous short stories to come out of this time period was Ward No. 6 by Anton Chekhov.

http://en.wikipedia.org/wiki/Short_story#Modern_short_stories

[Borges]

There should be a distinction made between the short story and the 'sketch'. Technically there were what you could call short stories in Classical Greece. Boccaccio and Chaucer were following up on that tradition as was someone like Washington Irving. The Checkovian short story probably has its roots in Maupassant more the in Poe.