Reference no: EM13990653
4. If (X, T ) and (Y, U ) are topological spaces, A is a base for T and B is a base for U , show that the collection of all sets A × B for A ∈ A and B ∈ B is a base for the product topology on X × Y.
5. (a) Prove that any intersection of topologies on a set is a topology.
(b) Prove that for any collection U of subsets of a set X , there is a smallest topology on X including U , using part (a) (rather than subbases).
6. (a) Let An be the set of all integers greater than n. Let Bn be the collection of all subsets of {1, ..., n}. Let Tn be the collection of sets of positive integers that are either in Bn or of the form An ∪ B for some B ∈ Bn . Prove that Tn is a topology.
(b) Show that Tn for n = 1, 2,... , is an inclusion-chain of topologies whose union is not a topology.
(c) Describe the smallest topology which includes Tn for all n.
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Describe the difference between mentoring and coaching
: Describe the difference between mentoring and coaching - Identify your customers and improve your understanding of their needs, what are 5 questions you should ask yourself?
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Inverse of the distribution function
: 1. Show that if S is the real line R with usual metric, then Theorem 11.7.2 holds for the probability space (0, 1) with Lebesgue measure. Hint: Let Xn be the inverse of the distribution function of Pn as defined in Proposi- tion 9.1.2.
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How many people can be lifted one story with a megajoule
: What is the force of gravity on a 1 kg mass ? What energy is needed to lift a 1 kg mass a distance of 3 meters ? What energy is needed to lift you one story in this building ?
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Convergence for the product topology
: This situation comes up especially when the Xi are [copies of] the same space, such as R with its usual topology. Then ITi Xi is the set of all functions from I into R, often called RI .)
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Problem regarding the inclusion-chain of topologies
: Let An be the set of all integers greater than n. Let Bn be the collection of all subsets of {1, ..., n}. Let Tn be the collection of sets of positive integers that are either in Bn or of the form An ∪ B for some B ∈ Bn . Prove that Tn is a topol..
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How high can the person climb with this energy
: A calorie is the amount of heat energy needed to raise the temperature of 1 gram of water 1 degree Celsius. (Heat is proportional to the change in Temperature) The calorie is equal to 4.2 Joules. The food Calorie is 1000 times bigger (a kilocalori..
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Used to calculate the weighted-average cost of capital
: A company may have two weighted-average costs of capital if the firm's capital structure is so large that new common stock must be sold.
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Product of countably many separable topological spaces
: Show that the product of countably many separable topological spaces, with product topology, is separable.
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Ultrametric space and d an ultrametric
: For any two real numbers u and v, max(u, v) := u iff u ≥ v; otherwise, max(u, v) := v. A metric space (S, d) is called an ultrametric space and d an ultrametric if d(x, z) ≤ max(d(x, y), d(y, z)) for all x, y, and z in R. Show that in an ultrametr..
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