Geometry Unit 18: Euclidean vs Non-Euclidean Geometry 2009-2010
... and be different sizes. Two figures can be the same size and be different shapes. Two figures can have the same size only if they are the same shape. ...
... and be different sizes. Two figures can be the same size and be different shapes. Two figures can have the same size only if they are the same shape. ...
Algebraic topology exam
... 2. A) State and prove the Mayer-Vietoris theorem for singular or simplicial (nonreduced) homology. B) Show that if S(X) is the suspension of a topological space X, then Hp(S(X)) and Hp-1(X) are isomorphic in reduced homology. 3. A) Let K, L be simplicial complexes and f,g : |K| |L| homotopic maps. ...
... 2. A) State and prove the Mayer-Vietoris theorem for singular or simplicial (nonreduced) homology. B) Show that if S(X) is the suspension of a topological space X, then Hp(S(X)) and Hp-1(X) are isomorphic in reduced homology. 3. A) Let K, L be simplicial complexes and f,g : |K| |L| homotopic maps. ...
Math 4853 homework 29. (3/12) Let X be a topological space
... 29. (3/12) Let X be a topological space. Suppose that B is a collection of subsets of X such that (1) The sets in B are open in X. (2) For every open set U in X and every x ∈ U , there exists B ∈ B such that x ∈ B ⊆ U. (a) Prove that B is a basis. (b) Prove that the topology {V ⊆ X | ∀x ∈ V, ∃B ∈ B, ...
... 29. (3/12) Let X be a topological space. Suppose that B is a collection of subsets of X such that (1) The sets in B are open in X. (2) For every open set U in X and every x ∈ U , there exists B ∈ B such that x ∈ B ⊆ U. (a) Prove that B is a basis. (b) Prove that the topology {V ⊆ X | ∀x ∈ V, ∃B ∈ B, ...
Compactness of a Topological Space Via Subbase Covers
... of the product space ι∈I Xι is cruder than (i.e. is a subset of) the topology of the product of discrete topological spaces on the sets Xι ; since (we are assuming that) the latter topology is compact, the topology O is also compact. For a start, observe that the exercise becomes trivial if “subbase ...
... of the product space ι∈I Xι is cruder than (i.e. is a subset of) the topology of the product of discrete topological spaces on the sets Xι ; since (we are assuming that) the latter topology is compact, the topology O is also compact. For a start, observe that the exercise becomes trivial if “subbase ...
The Cantor Discontinuum
... 2. Every nonempty perfect set in R has the same cardinality as R. In particular, if D is a Cantor discontinuum and if D ⊂ R, then card D = card R = 2ℵ0 . 3. Cantor’s Middle Third Set K ⊂ [0, 1] ⊂ R has Lebesgue measure zero. 4. There exist Cantor discontinuums in R of positive Lebesgue measure. In f ...
... 2. Every nonempty perfect set in R has the same cardinality as R. In particular, if D is a Cantor discontinuum and if D ⊂ R, then card D = card R = 2ℵ0 . 3. Cantor’s Middle Third Set K ⊂ [0, 1] ⊂ R has Lebesgue measure zero. 4. There exist Cantor discontinuums in R of positive Lebesgue measure. In f ...
Section 9.4 - Geometry in Three Dimensions
... • A simple closed surface has exactly one interior, has no holes, and is hollow. It separates space into interior, surface, and exterior. • A sphere is the set of all points at a given distance from a given point, the center. • A solid is the set of all points on a simple closed surface along with a ...
... • A simple closed surface has exactly one interior, has no holes, and is hollow. It separates space into interior, surface, and exterior. • A sphere is the set of all points at a given distance from a given point, the center. • A solid is the set of all points on a simple closed surface along with a ...
ON DOUBLE-DERIVED SETS IN TOPOLOGICAL SPACES In [1
... 1. There is a subset A ⊂ X such that the double-derived set (Ad )d is not closed in X. 2. There is a subset A ⊂ X such that none of the iterated derived sets Ad , (Ad )d , ((Ad )d )d , . . . is closed. 3. X contains a copy of two-point indiscrete space, as a subset of the form F ∩ G with F closed an ...
... 1. There is a subset A ⊂ X such that the double-derived set (Ad )d is not closed in X. 2. There is a subset A ⊂ X such that none of the iterated derived sets Ad , (Ad )d , ((Ad )d )d , . . . is closed. 3. X contains a copy of two-point indiscrete space, as a subset of the form F ∩ G with F closed an ...
MATH 358 – FINAL EXAM REVIEW The following is
... • The winding number of f : [a, b] → S 1 . . . • A retract A of X. . . 2. True or False? Determine if each of the following statements are true of false. If a statement is true, give a short sketch of the proof; if false, provide a counter-example. All sets X, Y , etc. are assumed to be topological ...
... • The winding number of f : [a, b] → S 1 . . . • A retract A of X. . . 2. True or False? Determine if each of the following statements are true of false. If a statement is true, give a short sketch of the proof; if false, provide a counter-example. All sets X, Y , etc. are assumed to be topological ...
UNIT PLAN - Connecticut Core Standards
... formulas for surface area and volume. Later in the unit students study Cavalieri’s principle and encounter more formal derivations of the formulas for volume of the pyramid, cone and sphere. The unit explores several real world applications of surface area and volume as well as the geometry of the s ...
... formulas for surface area and volume. Later in the unit students study Cavalieri’s principle and encounter more formal derivations of the formulas for volume of the pyramid, cone and sphere. The unit explores several real world applications of surface area and volume as well as the geometry of the s ...
