Topology MA Comprehensive Exam
... quotient space are X/G connected then X is connected. 6. A topological space X is said to be locally connected if the connected components of each point form a base of neighborhoods of X. Prove that in a locally connected space the connected components of X are both closed and open in X. 7. Give the ...
... quotient space are X/G connected then X is connected. 6. A topological space X is said to be locally connected if the connected components of each point form a base of neighborhoods of X. Prove that in a locally connected space the connected components of X are both closed and open in X. 7. Give the ...
Problems for the exam
... and P2 is the plane P2 = {(x, y, z, w)|z = w = 0}. 8. Is it possible to realize CP2 as a finite CW-complex with an even number of cells in every dimension? 9. Viewing S 1 ⊂ C2 as the unit complex numbers, define a continuous map φ : S1 × S1 → S1 × S1 by φ(ξ1 , ξ2 ) = (ξ1 , ξ1 ξ2 ). Is φ homotopic to ...
... and P2 is the plane P2 = {(x, y, z, w)|z = w = 0}. 8. Is it possible to realize CP2 as a finite CW-complex with an even number of cells in every dimension? 9. Viewing S 1 ⊂ C2 as the unit complex numbers, define a continuous map φ : S1 × S1 → S1 × S1 by φ(ξ1 , ξ2 ) = (ξ1 , ξ1 ξ2 ). Is φ homotopic to ...
Surface Areas and Volumes of Spheres
... You will end up with two “figure 8” pieces of material, as shown above. From the amount of material it takes to cover the ball, what would you estimate the surface area S of the ball to be? Express your answer in terms of the radius r of the ball. ...
... You will end up with two “figure 8” pieces of material, as shown above. From the amount of material it takes to cover the ball, what would you estimate the surface area S of the ball to be? Express your answer in terms of the radius r of the ball. ...
June 2012
... 1) Assume that (X, τ ) is a topological space with the property that for every open set G ⊆ X, the closure of G, G, is open. Such topological spaces are called extremally disconnected. Prove the following. a) If F ⊆ X is a closed set, then the interior of F , F ◦ , is closed. b) If G ⊆ X is an open ...
... 1) Assume that (X, τ ) is a topological space with the property that for every open set G ⊆ X, the closure of G, G, is open. Such topological spaces are called extremally disconnected. Prove the following. a) If F ⊆ X is a closed set, then the interior of F , F ◦ , is closed. b) If G ⊆ X is an open ...
Math 106: Course Summary
... halves, keeping your finger on some point of the half-ball. Then the individual curvatures (coming from the slices) at the point of interest change but the Gauss curvature does not change. That is, the product of the extrema are constant. This fact, when formalized, is Gauss’ famous Theorema Egregiu ...
... halves, keeping your finger on some point of the half-ball. Then the individual curvatures (coming from the slices) at the point of interest change but the Gauss curvature does not change. That is, the product of the extrema are constant. This fact, when formalized, is Gauss’ famous Theorema Egregiu ...
finite intersection property
... T = {Cα }α∈J of closed subsets of X having the finite intersection property, α∈J Cα 6= ∅. An important special case of the preceding is that in which C is a countable collection of non-empty nested sets, i.e., when we have C1 ⊃ C2 ⊃ C3 ⊃ · · · . In this case, C automatically has the finite intersect ...
... T = {Cα }α∈J of closed subsets of X having the finite intersection property, α∈J Cα 6= ∅. An important special case of the preceding is that in which C is a countable collection of non-empty nested sets, i.e., when we have C1 ⊃ C2 ⊃ C3 ⊃ · · · . In this case, C automatically has the finite intersect ...
1 Practice Problems
... b Describe the compact sets in the lower limit topology on R. (The reals with the lower limit topology R` is often called the Sorgenfrey line.) c Is the Sorgenfrey line connected? Locally connected? Describe its components. Additional Problem 2. 1. Let X be a (nonempty) compact Hausdorff space. If e ...
... b Describe the compact sets in the lower limit topology on R. (The reals with the lower limit topology R` is often called the Sorgenfrey line.) c Is the Sorgenfrey line connected? Locally connected? Describe its components. Additional Problem 2. 1. Let X be a (nonempty) compact Hausdorff space. If e ...
SIMPLEST SINGULARITY IN NON-ALGEBRAIC
... space has to be Moishezon. For dimension 2, it is a classical result that it is also sufficient, provided X is non-singular (Chow and Kodaira, 1952). In general it is not clear how to determine algebraicity of normal (singular) Moishezon surfaces and our understanding of non-algebraic Moishezon surf ...
... space has to be Moishezon. For dimension 2, it is a classical result that it is also sufficient, provided X is non-singular (Chow and Kodaira, 1952). In general it is not clear how to determine algebraicity of normal (singular) Moishezon surfaces and our understanding of non-algebraic Moishezon surf ...
(bring lecture 3 notes to complete the discussion of area, perimeter
... Any point P on an angle bisector is equidistant from the sides of the angle. Any point that is equidistant from the sides of an angle is on the angle bisector of the angle. Any point equidistant from the endpoints of a segment is on the perpendicular bisector of the segment. Any point on the perpend ...
... Any point P on an angle bisector is equidistant from the sides of the angle. Any point that is equidistant from the sides of an angle is on the angle bisector of the angle. Any point equidistant from the endpoints of a segment is on the perpendicular bisector of the segment. Any point on the perpend ...