Re: metric spaces

On Sat, 15 Jul 2006, William Elliot wrote:

Typo *** corrected at end of this post.

On Sat, 15 Jul 2006, bill wrote:

(X, d) is an arbitrary metric space, and Y = R with d2.

What you mean
with d2?

f : X => R is continuous on X.

Typo, X -> R. I'll consider R to the the reals with the usual
open interval topology.

Can we show that for every c in R, the set {x in X : f(x) > c} is an
open set in X?

Yes. (c,oo) is an open set and since f is continuous
{ x | c < f(x) } = f^-1((0,oo))

*** { x | c < f(x) } = f^-1((c,oo))

is open.

.

Relevant Pages

  • Re: proving something is a semi algebra.
    ... element form a semi algebra, i have proven all the properties bar ... It could be a typo, ... im thinking with the sets in the form [a,b) is not compact at i ... cannot represent the end point b in an open set. ...
    (sci.math)
  • Re: proving something is a semi algebra.
    ... element form a semi algebra, i have proven all the properties bar ... It could be a typo, ... im thinking with the sets in the form [a,b) is not compact at i ... cannot represent the end point b in an open set. ...
    (sci.math)
  • Re: metric spaces
    ... On Sat, 15 Jul 2006, bill wrote: ... Typo, X -> R. I'll consider R to the the reals with the usual ... is an open set and since f is continuous ...
    (sci.math)
  • Re: 1 to 10
    ... William Elliot writes: ... Surely that last line is a typo? ... parentheses: 1111!,!,!)!, and so on. ... No biggest. ...
    (sci.math)
  • Re: Multiplicative group of units (Z_p^2)^*
    ... > Newsgroups: sci.math ... > William Elliot wrote: ... I typo I guess. ... > So for this to be of any possible use, first find, for any generator ...
    (sci.math)