# Algebraic theories by Wraith, Gavin

By Wraith, Gavin

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The interior ofA is the largest open set containedin A. ) Moreover, a point p € A is an interior point of A if and only if there exists an open neighborhood N ofp such that N C A. Proof If O is any open set contained in A then, by Definition 20, O C A°. 22 Suppose p G A0. Then A0 is the required open neighborhood of p satisfying A° C A. If p has an open neighborhood that is contained in A, then p is an interior point directly from Definition 20. D As an intuitive example, let A = Br(x) in R n , and then A° = Br(x).

Then Be{x) C O. This shows that any finite intersection of open sets is open. Trivially, B\{x) C R n for any x e R n so R n itself is open. Since there are no points in the empty set 0 , for every i E 0 there is an open ball Be{x) C 0 because this is a vacuous statement. When one encounters a new topological space it is good practice to verify that the open sets satisfy the axioms from Definition 7. Now that we have the definition of a topological space, we can state the definition of continuity for general topological spaces.

Proof. Define the map / : Q -> Z x Z by /(f) = <*«)■ where p / g is written in reduced form. This map is a one-to-one correspondence between Q and a subset of a countable set, and hence Q is countable. Since Q contans Z, it is infinite. ) Hence Q is countably infinite. D PROPOSITION 4. The set R is uncountable. Proof Suppose, to obtain a contradiction, that R is countable. Then the interval (0,1) is countable. }. ) a£ + 1 mod 10. Then x is an element of (0,1) which is not equal to any of the α$.