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:
“It would be disingenuous, however, not to point out that some things are considered as morally certain, that is, as having sufficient certainty for application to ordinary life, even though they may be uncertain in relation to the absolute power of God.”
—René Descartes (1596–1650)
“Science is the language of the temporal world; love is that of the spiritual world. Man, indeed, describes more than he explains; while the angelic spirit sees and understands. Science saddens man; love enraptures the angel; science is still seeking, love has found. Man judges of nature in relation to itself; the angelic spirit judges of it in relation to heaven. In short to the spirits everything speaks.”
—Honoré De Balzac (1799–1850)
“There is the falsely mystical view of art that assumes a kind of supernatural inspiration, a possession by universal forces unrelated to questions of power and privilege or the artist’s relation to bread and blood. In this view, the channel of art can only become clogged and misdirected by the artist’s concern with merely temporary and local disturbances. The song is higher than the struggle.”
—Adrienne Rich (b. 1929)
“It is an axiom in political science that unless a people are educated and enlightened it is idle to expect the continuance of civil liberty or the capacity for self-government.”
—Texas Declaration of Independence (March 2, 1836)
“We must hold a man amenable to reason for the choice of his daily craft or profession. It is not an excuse any longer for his deeds that they are the custom of his trade. What business has he with an evil trade?”
—Ralph Waldo Emerson (1803–1882)