## Low-dimensional geometry: From Euclidean surfaces to by Francis Bonahon

By Francis Bonahon

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Low-dimensional geometry: From Euclidean surfaces to hyperbolic knots

The learn of three-dimensional areas brings jointly parts from a number of components of arithmetic. the main awesome are topology and geometry, yet parts of quantity concept and research additionally make appearances. some time past 30 years, there were outstanding advancements within the arithmetic of three-d manifolds.

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Proof (i) ⇒ (ii): Let x = y in X; by hypothesis, there exist disjoint neighbourhoods U of x, V of y. Since a net cannot eventually belong to each of two disjoint sets, it is clear that no net in X can converge to both x and y. (ii) ⇒ (i): Suppose that (X, T ) is not Hausdorff and that x = y are points in X for which every neighbourhood of x intersects every neighbourhood of y. Let Nx (Ny ) be the neighbourhood systems at x (y) respectively. Then both Nx and Ny are directed by reverse inclusion.

Again, f (x) > α iff there exist r1 , r2 such that α < r1 < r2 < f (x), implying that x ∈ Gr2 whence x ∈ G¯r1 . It follows that f −1 ((α, 1]) = ∪r1 >α (X \ G¯r1 ), which is again open. 4 Every T4 space is T3 1 . 1. 8 Any compact T2 space is T4 . 3. Note Unlike the previous axioms, T4 is neither hereditary nor productive. The global view of the hierarchy can now be filled in as an exercise from data supplied above:Metrizable Hereditary? Productive? T4 T3 1 2 T3 T2 T1 The following is presented as an indication of how close we are to having ‘come full circle’.

I−1 k k we can replace each by (G ∩ H) ∩ . . . ∩ πi−1 k and thus eliminate all repetitions. It is routine to check that if Tn is the usual topology on Rn , and T the usual topology on R, then (R, T ) × (R, T ) × . . (R, T ) = (Rn , Tn ) as one would hope! 1 In a product space (X, T ), N is a neighbourhood of p ∈ X iff there exists some open box B such that p ∈ B ⊆ N . 2 For each i ∈ I, (i) πi is continuous (ii) πi is an open mapping. Proof (i) Immediate. (ii) Given open G ⊆ X, then G is a union of basic open sets {Bk : k ∈ K} in X, whence πi (G) is a union of open subsets {Bki : k ∈ K} of Xi and is therefore open.