Math 525 Notes for sec 22 Final Topologies Let Y be a set, {(X i,τi
... space is homeomorphic to a circle, considered as a subspace of R2 . Example 2 Let (X, τ ) be the closed unit square [0, 1] × [0, 1], considered as a subspace of R2 . We define a quotient set X̂ by identifying pairs of points on the two vertical boundaries which are the same height above the x-axis ( ...
... space is homeomorphic to a circle, considered as a subspace of R2 . Example 2 Let (X, τ ) be the closed unit square [0, 1] × [0, 1], considered as a subspace of R2 . We define a quotient set X̂ by identifying pairs of points on the two vertical boundaries which are the same height above the x-axis ( ...
HOMEWORK MATH 445 11/7/14 (1) Let T be a topology for R
... (1) Let X be a compact Hausdorff topological space. Let A, B be disjoint closed subsets of X. Prove that there exist disjoint open sets U , V such that A ⊆ U and B ⊆ V . ...
... (1) Let X be a compact Hausdorff topological space. Let A, B be disjoint closed subsets of X. Prove that there exist disjoint open sets U , V such that A ⊆ U and B ⊆ V . ...
BASIC ALGEBRAIC TOPOLOGY: THE FUNDAMENTAL GROUP OF
... Example 1.4. In R , any two paths f0 and f1 that have the same endpoints are homotopic via the linear homotopoy defined by ft (s) = (1 − t)f0 (s) + tf1 (s). This means that each f0 (s) travels along the line segments to f1 (s) at a constant speed. Proposition 1.5. Given a topological space X with tw ...
... Example 1.4. In R , any two paths f0 and f1 that have the same endpoints are homotopic via the linear homotopoy defined by ft (s) = (1 − t)f0 (s) + tf1 (s). This means that each f0 (s) travels along the line segments to f1 (s) at a constant speed. Proposition 1.5. Given a topological space X with tw ...
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.