Heine-Borel TheoremClosed and bounded inRnis equivalent to compact. Proof: AssumeEis compact, which means that every open cover of it has a finite subcover. Apply this to any arbitrary covering ofEby rational open balls, and since a finite subcover exists, the subcover is bounded. Suppose b...
We will consider four formal spaces, or locales, namely formal Cantor spaceC, formal Baire spaceB, the formal real lineR, and the formal function spaceRRbeing the exponential in the category of locales (cf. [3]). The corresponding spaces of points will be denoted by pt(C), pt(B), pt...