Lecture 5 and 6
... d) for all U ∈ U0 and x ∈ E there is an ε > 0, so that λx ∈ U , for all λ ∈ K with |λ| < ε , e) if (E, T ) is Hausdorff, then for every x ∈ E, x (= 0, there is a U ∈ U0 with x (∈ U , f ) if E is locally convex, then there is for all U ∈ U0 a convex V ∈ U0 , with V ⊂ U , i.e. 0 has a neighborhood bas ...
... d) for all U ∈ U0 and x ∈ E there is an ε > 0, so that λx ∈ U , for all λ ∈ K with |λ| < ε , e) if (E, T ) is Hausdorff, then for every x ∈ E, x (= 0, there is a U ∈ U0 with x (∈ U , f ) if E is locally convex, then there is for all U ∈ U0 a convex V ∈ U0 , with V ⊂ U , i.e. 0 has a neighborhood bas ...
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.