Hyperboloid Is (Path) Connected

Question

Is hyperboloid $x^2+y^2-z^2=1$ connected? From the picture it is connected but I don't know how to formalise this in the language of topology. I'm also wondering whether it is path-connected or not since the connected component hyperbola is path connected. Thank you.

Answer 1

Yes, this surface is connected and path-connected. Path-connected implies connected, so it suffices to show path-connectedness.

To show a space is path-connected, you don't need to directly construct a path between an arbitrary pair of points, you just need to show that the equivalence relation $\sim$ defined by "$x \sim y$ iff there is a path from $x$ to $y$" has exactly one equivalence class.

So, hint: find a path $P$ through this hyperboloid such that every point on the hyperboloid has an easily constructed path to some point in $P$.

If you can do this, you'll have shown that every point in the hyperboloid lies in the equivalence class of some point in $P$, but $P$ itself is a path, so all of the points in $P$ lie in the same equivalence class! Done.

Please give this a shot and comment if you're still stuck!

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A different technique would be to show that this surface is homeomorphic to $S^1 \times \mathbb{R}$, and then apply the general fact that a product of path-connected spaces is path-connected.