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)
“You see, I am alive, I am alive
I stand in good relation to the earth
I stand in good relation to the gods
I stand in good relation to all that is beautiful
I stand in good relation to the daughter of Tsen-tainte
You see, I am alive, I am alive”
—N. Scott Momaday (b. 1934)
“The whole point of Camp is to dethrone the serious. Camp is playful, anti-serious. More precisely, Camp involves a new, more complex relation to “the serious.” One can be serious about the frivolous, frivolous about the serious.”
—Susan Sontag (b. 1933)
“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)
“A girl loves most often because she is loved,Mnot from choice on her part. She is won by the flattery of the man’s desire.”
—Anthony Trollope (1815–1882)