notes on the proof Tychonoff`s theorem
... of C, then U := D is an upper bound for D. To see this, we need to show that U has is a cover with no finite subcover. It clearly is a cover. If it had a finite subcover V = {V1, . . . , Vn } ⇢ U, then since there are only finitely many Vi they must be all contained in a fixed U ⇢ D which would then ...
... of C, then U := D is an upper bound for D. To see this, we need to show that U has is a cover with no finite subcover. It clearly is a cover. If it had a finite subcover V = {V1, . . . , Vn } ⇢ U, then since there are only finitely many Vi they must be all contained in a fixed U ⇢ D which would then ...
A SEPARATION AXIOM BETWEEN SEMI-To AND
... Theorem 2.9 A semi-T0 space X is semi-D 1, iffthere is not sc.c point in X. Proof. Necessity. If X is semi-D 1 then each point xEX, belongs toa sD-sets S= 0 1 \ 0 2 , hence xE 0 1 . Since 0 1 -:t= X, x is not se. e point. Sufficiency. If X is semi-T0 , then for each distinct pair of points x,yEX, at ...
... Theorem 2.9 A semi-T0 space X is semi-D 1, iffthere is not sc.c point in X. Proof. Necessity. If X is semi-D 1 then each point xEX, belongs toa sD-sets S= 0 1 \ 0 2 , hence xE 0 1 . Since 0 1 -:t= X, x is not se. e point. Sufficiency. If X is semi-T0 , then for each distinct pair of points x,yEX, at ...
on geometry of convex ideal polyhedra in hyperbolic
... to a complete hyperbolic surface F, immersed in H3. F is topologically equivalent to an infinite cylinder, and both its ends are of infinite volume. (The first part is clear, since each completed face x is a topologically an infinite strip. Geometrically, the “left” (orienting F” in a consistent fas ...
... to a complete hyperbolic surface F, immersed in H3. F is topologically equivalent to an infinite cylinder, and both its ends are of infinite volume. (The first part is clear, since each completed face x is a topologically an infinite strip. Geometrically, the “left” (orienting F” in a consistent fas ...
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. Intuitively, a 3-manifold can be thought of as a possible shape of the universe. Just like a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.