The Urysohn Metrization Theorem
... Let {Bn } be a countable basis for X. For each pair of n, m indices for which B¯n ⊂ Bm , apply the Urysohn lemma to choose a continuous function gn,m : X → [0, 1] such that gn,m (B¯n ) = {1} and gn,m (X −Bm ) = {0}. Then the collection gn,m satisfies our requirement: Given x0 and given open set U of ...
... Let {Bn } be a countable basis for X. For each pair of n, m indices for which B¯n ⊂ Bm , apply the Urysohn lemma to choose a continuous function gn,m : X → [0, 1] such that gn,m (B¯n ) = {1} and gn,m (X −Bm ) = {0}. Then the collection gn,m satisfies our requirement: Given x0 and given open set U of ...
GCH2L1
... Determine if each conjecture is true. If false, give a counterexample. 3. The quotient of two negative numbers is a positive number. true 4. Every prime number is odd. false; 2 false; 90° and 90° 5. Two supplementary angles are not congruent. 6. The square of an odd integer is odd. true Holt Geometr ...
... Determine if each conjecture is true. If false, give a counterexample. 3. The quotient of two negative numbers is a positive number. true 4. Every prime number is odd. false; 2 false; 90° and 90° 5. Two supplementary angles are not congruent. 6. The square of an odd integer is odd. true Holt Geometr ...
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.