In-class solutions. - Dartmouth Math Home
... (b) What is the smallest topology on X containing both τ1 and τ2 ? Such a topology τ must contain τ1 ∪ τ2 = {X, ∅, {a}, {b}, {c}, {a, b}, {a, c}}. As a topology is contained under unions, we must have {b} ∪ {c} = {b, c} ∈ τ . This compels τ = P(X), the discrete topology. (Note: you did not need to j ...
... (b) What is the smallest topology on X containing both τ1 and τ2 ? Such a topology τ must contain τ1 ∪ τ2 = {X, ∅, {a}, {b}, {c}, {a, b}, {a, c}}. As a topology is contained under unions, we must have {b} ∪ {c} = {b, c} ∈ τ . This compels τ = P(X), the discrete topology. (Note: you did not need to j ...
Discrete and Continuous Random Variables
... conditions, is 60 kN. Three piles are selected randomly for testing. Let X be the number of piles having strength under 60 kN, from amongst the three selected. The random variable X can have value 0, 1, 2, or 3. If S is the event that a pile has strength exceeding 60 kN, and F is the event that its ...
... conditions, is 60 kN. Three piles are selected randomly for testing. Let X be the number of piles having strength under 60 kN, from amongst the three selected. The random variable X can have value 0, 1, 2, or 3. If S is the event that a pile has strength exceeding 60 kN, and F is the event that its ...
6 | Continuous Functions
... 6.12 Note. If � : X → Y is a continuous bijection then � need not be a homeomorphism since the inverse function � −1 may be not continuous. For example, let X = {�1 � �2 } be a space with the discrete topology and let Y = {�1 � �2 } be a space with the antidiscrete topology. Let � : X → Y be given b ...
... 6.12 Note. If � : X → Y is a continuous bijection then � need not be a homeomorphism since the inverse function � −1 may be not continuous. For example, let X = {�1 � �2 } be a space with the discrete topology and let Y = {�1 � �2 } be a space with the antidiscrete topology. Let � : X → Y be given b ...
MATH0055 2. 1. (a) What is a topological space? (b) What is the
... (b) What is the discrete topology on a set? Let X be a finite set and T a topology on X for which all singleton sets {x}, x ∈ X, are closed. Show that T is the discrete topology. Does the same conclusion hold if X is countable? Give a proof or find a counterexample. ...
... (b) What is the discrete topology on a set? Let X be a finite set and T a topology on X for which all singleton sets {x}, x ∈ X, are closed. Show that T is the discrete topology. Does the same conclusion hold if X is countable? Give a proof or find a counterexample. ...
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... Now expand this as a power series. Given a partition of n with ai parts of size i ≥ 1, we get a term xn in this expansion by choosing xa1 from the first term in the product, x2a2 from the second, x3a3 from the third and so on. Clearly any term xn in the expansion arises in this way from a partition ...
... Now expand this as a power series. Given a partition of n with ai parts of size i ≥ 1, we get a term xn in this expansion by choosing xa1 from the first term in the product, x2a2 from the second, x3a3 from the third and so on. Clearly any term xn in the expansion arises in this way from a partition ...
