[chacham15]

This still doesnt make sense to me. Imagine a continuous line whose positions map to the real numbers between 0 and 1. If I "move" the line over 0.1 wrapping the end back to the beginning (i.e. x2 = (x1 + 0.1) % 1), there will be no points that are in the same position as they were in before.

EDIT: If you need a continuous function, wouldnt expanding the space to a line from -Inf to +Inf and then using x2 = x1 + 0.1 do the trick?

[contravariant]

Brouwer's fixed point theorem only applies to compact convex sets.

Infinite lines don't work, as they are not compact.

Similarly a circle would not work as it is not convex (you're close with your example, you just need to glue together the endpoints to turn it into a circle and make the map continuous).

[d_tr]

The line example is what I thought and then I looked the theorem up in Wikipedia. Thanks for pointing this out.

[frutiger]

The theorem only applies to continuous functions.