MANIFOLDS AND CONNECTEDNESS Proposition 1. Let X be a
... Proposition 1. Let X be a topological manifold. Then X is locally connected. In other words, for every x ∈ X, x ∈ U , U open in X, there is a connected open set V so that x ∈ V ⊂ U . Remark. This is not true for all topological spaces Y . An example is Y = {0, 1, 1/2, 1/3, 1/4, . . . } with the subs ...
... Proposition 1. Let X be a topological manifold. Then X is locally connected. In other words, for every x ∈ X, x ∈ U , U open in X, there is a connected open set V so that x ∈ V ⊂ U . Remark. This is not true for all topological spaces Y . An example is Y = {0, 1, 1/2, 1/3, 1/4, . . . } with the subs ...
HOMEOMORPHISM GROUPS AND THE TOPOLOGIST`S SINE
... / K there is a positive ε so that [0, ε] × [1 − ε] is disjoint from K. Select a natural number N such that sin(π/2−2−N ) > 1−ε and 1/2πN is less than both d and ε. Let n ≥ N and consider hn and a point (x, y) in K. If (x, y) ∈ A or x ≥ 1/(rn − 2−n ), then hn (x, y) = (x, y) ∈ O. If 0 < x ≤ 1/(rn + 2 ...
... / K there is a positive ε so that [0, ε] × [1 − ε] is disjoint from K. Select a natural number N such that sin(π/2−2−N ) > 1−ε and 1/2πN is less than both d and ε. Let n ≥ N and consider hn and a point (x, y) in K. If (x, y) ∈ A or x ≥ 1/(rn − 2−n ), then hn (x, y) = (x, y) ∈ O. If 0 < x ≤ 1/(rn + 2 ...
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.