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I am asked whether the unit sphere $$X=\{(x,y,z,w)|x^2 + y^2 + z^2 + w^2 = 1 \}\subset \mathbb{R}^4,$$ is path connected or not.

I just know that $X$ is a closed subset. How can I answer this question?

Is there any hint? Thank you very much.

Servaes
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user115608
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1 Answers1

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The set $X$ is the unit sphere in $\Bbb{R}^4$. Any two distinct $x,y\in\Bbb{R}^4$ are contained in some plane $Y\subset\Bbb{R}^4$, and hence $x,y\in X\cap Y$. But $X\cap Y$ is then a circle in $Y$, which is of course path connected. So $x$ and $y$ are connected by a path in $X\cap Y\subset X$, hence $X$ is path connected.

Servaes
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  • thanks!that's true.but is there any method for such problems?for example how about ${(x,y,z)| x^2 + y^2 − z^2 = 1}\subset R^3$? – user115608 May 27 '16 at 10:31
  • I'm not familiar with any general methods. Subsets of Euclidean space in general can be very nasty... But drawing pictures often helps. – Servaes May 27 '16 at 11:18
  • ok.thanks.do u have any idea for this special case i mentioned? – user115608 May 27 '16 at 11:19
  • I would also suggest drawing a picture, and considering lower-dimensional slices. Consider asking a new question if you want a more elaborate answer. – Servaes May 27 '16 at 11:22
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    *In this case, slices with constant $z$-value are circles and slices with constant $x$- or $y$-values are hyperbolas. Every point is on such a circle, and every pair of circles is joined by a hyperbola, so every pair of points is joined by a path. – Servaes May 27 '16 at 11:25