Talk:Cross-cap

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The last sentence of the first paragraph is false. How should I fix this problem? sam Mon Aug 16 16:58:40 CDT 2004

This is referred to in Möbius strip: if you understand what's written there, try copying it over here :-) --Phil | Talk 08:01, Nov 4, 2004 (UTC)

[edit] Accuracy of the article

There is virtually nothing in the article (other than the lovely graphics) that is not false. Starting with the misguided attempt to roughly define a cross-cap in the first sentence.

• First of all, a cross-cap is never topologically equivalent to a Moebius strip. It is a continuous image by a certain type of map of the (closed) Moebius strip into 3-space, a map that has an open interval's worth of double-points.

• As a topological subspace of 3-space, it is the space obtained by starting with a closed disk D², choosing an interval in the disk's interior -- say the image of [-1,1] via an embedding h: [-1,1] → int(D²) -- and then identifying the points h(x) and h(-x) with each other for all x in (0,1].

• A cross-cap may have a boundary that is a round (perfect) circle, but is required only to have a boundary that is an unknotted simple closed curve.

• Further, there are continuous deformations of the usual picture of a Moebius band to a picture that is still topologically a Moebius band embedded in 3-space (i.e., with no self-intersections), such that its boundary is a perfect circle. So this perfect-circle-boundary property in no way characterizes a cross-cap.

• The word cross-cap has been erroneously used to mean a cross-cap with a disk glued on to its boundary (making a continuous image of a projective plane). But this is an error, and it should not be perpetuated in this article. Rather, the article should warn people to avoid this misuse of the word.Daqu 00:53, 4 December 2006 (UTC)Daqu 01:18, 4 December 2006 (UTC)