HOMEOMORPHISM IN IDEL TOPOLOGICAL SPACES Author: N.CHANDRAMATHI , K. BHUVANESWARI S.BHARATHI, INDIA
... Theorem 2.4: For any bijection f:(X,τ ,I)→(Y,σ,J),the following statements are equivalent: (i) f−1 :(Y,σ,J)→(X,τ ,I) is ωI-continuous (ii) f is an ωI-open map. (iii)f is an ωI-closed map proof:(i)→(ii):Let U be an I-open set of (X,τ ,I). By assumption (f−1 )−1 =f(U) is ωI-open in (Y,σ) and so f is ω ...
... Theorem 2.4: For any bijection f:(X,τ ,I)→(Y,σ,J),the following statements are equivalent: (i) f−1 :(Y,σ,J)→(X,τ ,I) is ωI-continuous (ii) f is an ωI-open map. (iii)f is an ωI-closed map proof:(i)→(ii):Let U be an I-open set of (X,τ ,I). By assumption (f−1 )−1 =f(U) is ωI-open in (Y,σ) and so f is ω ...
Spring 1998
... [0, 1) and (0, 1] are homeomorphic via x 7→ 1 − x. Other pairs are not homeomorphic. For a topological space X denote by N (X) the set of its nonseparating points, i.e., points x0 ∈ X such that X r {x0 } is connected. Clearly, N (X) is a topological invariant. On the other hand, one has N (S 1 ) = S ...
... [0, 1) and (0, 1] are homeomorphic via x 7→ 1 − x. Other pairs are not homeomorphic. For a topological space X denote by N (X) the set of its nonseparating points, i.e., points x0 ∈ X such that X r {x0 } is connected. Clearly, N (X) is a topological invariant. On the other hand, one has N (S 1 ) = S ...
13b.pdf
... The tangent sphere bundle T S(M) is the fibration over M with fiber the sphere of rays through O in T (M). When M is Riemannian, this is identified with the unit tangent bundle T1 (M). Proposition 13.4.2. Let O be a two-orbifold. If O is elliptic, then T1 (O) is an elliptic three-orbifold. If O is E ...
... The tangent sphere bundle T S(M) is the fibration over M with fiber the sphere of rays through O in T (M). When M is Riemannian, this is identified with the unit tangent bundle T1 (M). Proposition 13.4.2. Let O be a two-orbifold. If O is elliptic, then T1 (O) is an elliptic three-orbifold. If O is E ...
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.