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Clash Royale CLAN TAG #URR8PPP up vote 3 down vote favorite I have read somewhere that one motivation of scheme theory is that using methods developed in scheme theory we can detect that the point associated to the intersection of $y = 0 cap y = x^2$ has multiplicity two. Can someone explain this in details to me. That would be very helpful. algebraic-geometry schemes share | cite | improve this question edited Jul 30 at 2:08 Eric Wofsey 162k 12 188 298 asked Jul 29 at 21:09 Newbie 384 1 11 2 $k[x,y]/ (y - x^2, y) = k[x,y] / (y,x^2) = k[x]/(x^2)$. The final ring is a vector space of dimension 2, which is the multiplicity of the intersection. – Lorenzo Jul 29 at 21:18 1 If you really want all the details, you might have to start studying scheme theory. I recommend "The geometry of schemes" , by Eisenbud and Harris. – Jesko Hüttenhain Jul 29 a...

Clash Royale CLAN TAG #URR8PPP up vote 3 down vote favorite In Michael Coornaert's book 'Topological Dimension and Dynamical Systems' he mentions the existence of a set $X$ in the plane that is totally separated, but not zero-dimensional. He says it is due to Sierpinski, but the paper he cites is in French. The example doesn't seem to be in Steen & Seebach; is anyone familiar? Here, 'totally separated' means that the quasicomponents of $X$ are points. Equivalently, for any $x, y in X$ there is a separation of $X$ into disjoint open sets $U, V$ such that $x in U$ and $y in V$. The Sierpinski paper is here: https://eudml.org/doc/212954 A zero-dimensional space in this context can equivalently mean a space with a clopen basis, a space all of whose points have a clopen basis, or a space with arbitrarily fine covers by disjoint open sets. EDIT: David Hartley below has described Sierpinski's space satisfying the desired requirements. I've atta...