Homework 5 - Department of Mathematics
... 1. Prove that R is homeomorphic to (0, 1) (put in all details). 2. Find a continuous function f : R → R which is not open. 3. Which of the topologies on the 3 point set found in HW 1, are homeomorphic? 4. Show that a function f : X → Y between metric spaces is continuous if and only if whenever a se ...
... 1. Prove that R is homeomorphic to (0, 1) (put in all details). 2. Find a continuous function f : R → R which is not open. 3. Which of the topologies on the 3 point set found in HW 1, are homeomorphic? 4. Show that a function f : X → Y between metric spaces is continuous if and only if whenever a se ...
Geometry 10.5 student copy
... The base of a regular pyramid is a regular polygon, and the lateral faces are congruent isosceles triangles. The slant height of a regular pyramid is the distance from the vertex to the midpoint of an edge of the base. The altitude of a pyramid is the perpendicular segment from the vertex to the pla ...
... The base of a regular pyramid is a regular polygon, and the lateral faces are congruent isosceles triangles. The slant height of a regular pyramid is the distance from the vertex to the midpoint of an edge of the base. The altitude of a pyramid is the perpendicular segment from the vertex to the pla ...
HOMEOMORPHISM GROUPS AND THE TOPOLOGIST`S SINE
... and we recall that the topology on H(X) is generated by the subbasis {[K, O] : K compact and O open in X}. If X is locally compact, then composition is continuous on H(X) and if X is compact, then also the inverse operation is continuous thus H(X) is a topological group; see Arens [1]. The Cantor se ...
... and we recall that the topology on H(X) is generated by the subbasis {[K, O] : K compact and O open in X}. If X is locally compact, then composition is continuous on H(X) and if X is compact, then also the inverse operation is continuous thus H(X) is a topological group; see Arens [1]. The Cantor se ...
Topology
... πX : X × Y → X and πY : X × Y → Y are given by (x, y) 7→ x and (x, y) 7→ y. Prove that a function f : A → X × Y is continuous if and only if πX ◦ f and πY ◦ f are continuous. (2) Suppose (X, d) is a metric space such that the set A := {d(x1 , x2 ) : xi ∈ X. i = 1, 2} is the closed interval [0, 1]. ( ...
... πX : X × Y → X and πY : X × Y → Y are given by (x, y) 7→ x and (x, y) 7→ y. Prove that a function f : A → X × Y is continuous if and only if πX ◦ f and πY ◦ f are continuous. (2) Suppose (X, d) is a metric space such that the set A := {d(x1 , x2 ) : xi ∈ X. i = 1, 2} is the closed interval [0, 1]. ( ...
Prof. Girardi The Circle Group T Definition of Topological Group A
... Let’s look at some nice properties of T. Consider the natural projection π : R T given by π (θ) = [θ]. Then π is continuous since if dR (xn , x) → 0 then dT ([xn ] , [x]) → 0. Following directly from the definition of the quotient topology is that π is an open mapping and that T is Hausdorff. T is ...
... Let’s look at some nice properties of T. Consider the natural projection π : R T given by π (θ) = [θ]. Then π is continuous since if dR (xn , x) → 0 then dT ([xn ] , [x]) → 0. Following directly from the definition of the quotient topology is that π is an open mapping and that T is Hausdorff. T is ...