Louis Auslander's An Account of the Theory of Crystallographic Groups PDF

By Louis Auslander

Lawsuits of the yankee Mathematical Society
Vol. sixteen, No. 6 (Dec., 1965), pp. 1230-1236
Published by way of: American Mathematical Society
DOI: 10.2307/2035904
Stable URL: http://www.jstor.org/stable/2035904
Page count number: 7

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Additional resources for An Account of the Theory of Crystallographic Groups

Example text

We now collect some properties of limits. 18 (Uniqueness of limits in metric spaces). Let (X, d) be a metric space. If xk → p and xk → q as k → ∞, then p = q. Proof. Let ε > 0. Since xk → p and xk → q as k → ∞, it follows that there are some numbers N1 , N2 such that d(xk , p) < ε for all k > N1 and such that d(xk , q) < ε for all k > N2 . Hence by the triangle inequality for all k > max{N1 , N2 } we have d(p, q) ≤ d(p, xk ) + d(xk , q) < 2ε. Since this conclusion is true for any ε > 0, it follows that d(p, q) = 0 and hence p = q.

Third, let A = {Aj : j ∈ J} ⊂ τi . Now A Aj =i(Aj ) = i(Aj ) i(Aj )⊂i( ⊂ A) i A ⊂ A. j∈J Thus A = i( A) ∈ τi . Next, intτi (A) = {U ∈ τi | U ⊂ A} = {i(B) | i(B) ⊂ A, B ⊂ X} . Here we see that i(A) ⊂ intτi (A). Moreover, if i(A) ⊂ i(B) ⊂ A then i(A) = i(i(A)) ⊂ i(i(B)) = i(B) ⊂ i(A). Hence i(A) = intτi (A). 14. Above we have seen how topologies and closure operators (or interior operators) on a set are in bijective correspondence. 8. 15. For each j ∈ J, let τj be a topology on X. Prove that τ = τj j∈J is a topology.

On the other hand, it may well be that a countably infinite intersection of open sets is not open. In a metric space (X, d), ∞ B1/k (x) = {x}. k=1 Now {x} ∈ τd if and only if {x} = Br (x) for some r > 0. 14 (Properties of closed sets). Let X be a topological space. We have the following properties of closed sets in X: (C1) (C2) (C3) ∅ and X are closed in X. The intersection of any collection of closed subsets of X is closed. The union of a finite collection of closed subsets of X is closed. Proof.

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An Account of the Theory of Crystallographic Groups by Louis Auslander

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