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June 6

Separated/connected equivalence

If A and B are disjoint from each other's closure, can they be separated by open disjoint sets? The converse is easy, but going from disjoint closure to open sets seems impossible. I don't need the answer, only a confirmation of whether it's right or wrong. ThanksStandard Oil (talk) 06:30, 6 June 2009 (UTC)[reply]

If two sets satisfy the hypothesis satisfied by A and B above, they are called "separated sets." A topological space is said to be completely normal, if any two separated sets in the topological space, can be separated by open sets (according to the Wikipedia article). Equivalently, a topological space is completely normal if every subspace of X is normal. Given the high probability that the context of which you speak above is that of the metric spaces, the statement which you assert is correct. In particular, metric spaces are completely normal since a subspace of a metric space is a metric space in its own right. However, note that this property is not satisfied by topological spaces in general. --PS T 09:30, 6 June 2009 (UTC)[reply]

Thanks, I figured a while later that non-normal spaces don't satisfy the property that I couldn't prove, but didn't know it had to be completely normal for the property to be true. I was aware that metric spaces are normal (and thus completely normal like you said), but I needed this for all topological spaces...Standard Oil (talk) 17:15, 6 June 2009 (UTC)[reply]

Spherical wedge

Hello. I recently created Spherical wedge. An image would really brighten and clarify it, I think. For the pleasure of it, and so I know where to go for next time, could someone forward me to an application I might use to construct a sphere and perform operations such as colouring and labelling on it? I'm grateful for any response. —Anonymous Dissident^Talk 10:19, 6 June 2009 (UTC)[reply]

POV-Ray works exceptionally well for these purposes. It's a small, free and easy to learn application. But beware, you'll have to write the scene down. There's no graphical interface. — Kieff | Talk 19:24, 6 June 2009 (UTC)[reply]

Slightly off topic, but the term wedge to me does not inherently mean a wedge meade with two great circles. Is there some historic reason why wedge here applies to that condition? What does one call a wedge made with (a) Two equal sized non-great intersecting circles and (b) any two intersecting circles? -- SGBailey (talk) 19:54, 6 June 2009 (UTC)[reply]

That's a good question. All I know is that all my sources, old and new, have always mentioned the intersection of two halfplanes. The real definition for a spherical wedge is a solid of revolution obtained by revolving a semicircle through only x degrees, rather than the full 360 to produce a sphere. I'll look into it some more... —Anonymous Dissident^Talk 22:36, 6 June 2009 (UTC)[reply]

Another thing: I'm fairly sure the exterior lune is a defining feature, so you couldn't have a wedge defined within the sphere, as far as I know. —Anonymous Dissident^Talk 22:56, 6 June 2009 (UTC)[reply]

That's a nice picture, with the wedge separated from the rest of the sphere. How about doing one like it for spherical cap, to replace the non-separated one that's there now? 207.241.239.70 (talk) 06:54, 11 June 2009 (UTC)[reply]