IOSR Journal of Mathematics (IOSR-JM)
... Proof: Let x and y be any two distinct points of X. Then f x and f y are different points of Y because f is injective. Since Y is Hausdorff, there exist disjoint open sets U and V in Y containing f x and f y respectively. Since f is continuous and U ∩ V = ϕ, f −1 U and f −1 V are disjoint open sets ...
... Proof: Let x and y be any two distinct points of X. Then f x and f y are different points of Y because f is injective. Since Y is Hausdorff, there exist disjoint open sets U and V in Y containing f x and f y respectively. Since f is continuous and U ∩ V = ϕ, f −1 U and f −1 V are disjoint open sets ...
2. Unit 2 conjectures.
... If two angles form a linear pair of angles, then _____________________________________________ If two angles are vertical angles, then _____________________________________________ If two parallel lines are cut by a transversal, then corresponding angles are ______________________, alternate interio ...
... If two angles form a linear pair of angles, then _____________________________________________ If two angles are vertical angles, then _____________________________________________ If two parallel lines are cut by a transversal, then corresponding angles are ______________________, alternate interio ...
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.