Relation To The Axiom of Choice
The proofs of BCT1 and BCT2 for arbitrary complete metric spaces require some form of the axiom of choice; and in fact BCT1 is equivalent over ZF to a weak form of the axiom of choice called the axiom of dependent choices.
The restricted form of the Baire category theorem in which the complete metric space is also assumed to be separable is provable in ZF with no additional choice principles. This restricted form applies in particular to the real line, the Baire space ωω, and the Cantor space 2ω.
Read more about this topic: Baire Category Theorem
Famous quotes containing the words relation to the, relation to, relation, axiom and/or choice:
“Much poetry seems to be aware of its situation in time and of its relation to the metronome, the clock, and the calendar. ... The season or month is there to be felt; the day is there to be seized. Poems beginning “When” are much more numerous than those beginning “Where” of “If.” As the meter is running, the recurrent message tapped out by the passing of measured time is mortality.”
—William Harmon (b. 1938)
“Whoever has a keen eye for profits, is blind in relation to his craft.”
—Sophocles (497–406/5 B.C.)
“The foregoing generations beheld God and nature face to face; we, through their eyes. Why should not we also enjoy an original relation to the universe? Why should not we have a poetry and philosophy of insight and not of tradition, and a religion by revelation to us, and not the history of theirs?”
—Ralph Waldo Emerson (1803–1882)
“It’s an old axiom of mine: marry your enemies and behead your friends.”
—Robert N. Lee. Rowland V. Lee. King Edward IV (Ian Hunter)
“Therefore is love said to be a child
Because in choice he is so oft beguiled.”
—William Shakespeare (1564–1616)