Talk:Topology/Metric Spaces

Properties of the Interior of a Set
The last item in the "Properties of the Interior of a Set" talks about "A" being "open", but this is before the section which defines what an open set is. What's worse, proof of this property is left as an exercise to the reader, but there's nothing to prove since this is exactly the definition of an open set that's introduced later on. Am I missing something or does this need fixing?
 * No your not missing anything. Even though the proof is painfully trivial, it still needs to be proved.  But your correct if follows immediately.  Something does need to be fixed, the concept of open should probably come before interior.  I won't have time to rework it but your welcome to try if you like. Thenub314 (talk) 06:33, 7 June 2011 (UTC)
 * If the concept of open set is defined in terms of int(A) = A, and you put it before the concept of interior, we'd end up with the same issue. Also, regarding the proof exercise: the property is the same as the definition of open set. There is nothing to prove. They only differ in notation (prose vs. <=>), but they say exactly the same. I'd suggest just removing that line: it doesn't provide any substantial information (open sets are covered afterwards anyway) and it can easily lead to confusion. 85.58.134.159 (discuss) 22:10, 28 April 2013 (UTC)

Open vs Closed
"In \R, under the regular metric, the only set that is both open and closed is \R". What about the empty set? SuneJ

Metric Space Example: Hölder's inequality
Two lines above where it says "This is called Hölder's inequality", there is a line that seems not to be a valid result of the previous line. I think to make it valid you would have to have c^p the sum of the absolute values of c k p. If this is the case, what is stated as Hölder's inequality was already given a couple lines earlier, with a and b instead of c and d. Mathocd (talk) 07:25, 8 August 2010 (UTC)

Internal points
Hello

I have seen these called interior points, not internal points. And, in Royden's Real Analysis, there is a separate definition of something called an internal point. It has to do with convexity and is related, I believe, but is different. Is "internal" a standard name or would interior be better?

Thanks

Open Ball / Graphics
The graphics in section "Open Ball" are nicely rounded, but is that really appropriate for a mathematical text? Is there any consensus here? --Jerome Baum (talk) 08:21, 24 November 2009 (UTC)


 * I agree. The graphics are incorrect and possibly misleading. The corners should be perfectly sharp.130.243.130.84 (talk) 10:20, 18 September 2010 (UTC)
 * I agree too. These must change - they are too misleading.  --JamesCrook (talk) 11:05, 9 January 2011 (UTC)

Uniform Convergence
I believe in the section on uniform convergence p(fa(x),fb(x))<ε should be d(fa(x),fb(x))<ε     ChasR (discuss • contribs) 05:40, 8 May 2011 (UTC)
 * I'm under the same assumption as ChasR, so I'll be making the change Lundburgerr (discuss • contribs) 16:15, 13 July 2011 (UTC)

union of open sets
does the proof of union of open sets depend on the axiom of choice, for non-counteable sets?