Quotient Spaces and Quotient Maps
... For each x ∈ X, let Gx = {g(x) | g ∈ G}. View each of these “orbit” sets as a single point in some new space X ∗ . 2. Definition of quotient space Suppose X is a topological space, and suppose we have some equivalence relation “∼” defined on X. Let X ∗ be the set of equivalence classes. We want to d ...
... For each x ∈ X, let Gx = {g(x) | g ∈ G}. View each of these “orbit” sets as a single point in some new space X ∗ . 2. Definition of quotient space Suppose X is a topological space, and suppose we have some equivalence relation “∼” defined on X. Let X ∗ be the set of equivalence classes. We want to d ...
remarks on locally closed sets
... 3) if A, B are open mod J , then A ∩ B , A ∪ B and X \ A are open mod J , 4) A ⊆ X is open mod J if and only if A = U ∪ N where U is open and N is nwd in (X, τ ) . In order to state our main result in this section we need some more definitions. A subset S of a space (X, τ ) is called semi-locally cl ...
... 3) if A, B are open mod J , then A ∩ B , A ∪ B and X \ A are open mod J , 4) A ⊆ X is open mod J if and only if A = U ∪ N where U is open and N is nwd in (X, τ ) . In order to state our main result in this section we need some more definitions. A subset S of a space (X, τ ) is called semi-locally cl ...
SOME GEOMETRIC PROPERTIES OF CLOSED SPACE CURVES
... oriented tangents to γ1 and γ2 take values in the interval [π/2 − ε, π]. If we reverse the orientation on γ2 , then the angles between the oriented tangents to γ1 and γ2 will be replaced by the supplementary angles and will take values in the interval [0, π/2 + ε]. Remarks. 1. Thus, for immersed o ...
... oriented tangents to γ1 and γ2 take values in the interval [π/2 − ε, π]. If we reverse the orientation on γ2 , then the angles between the oriented tangents to γ1 and γ2 will be replaced by the supplementary angles and will take values in the interval [0, π/2 + ε]. Remarks. 1. Thus, for immersed o ...
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.