Consider R with the cofinite topology. Is [0,1] compact? Can you describe the compact sets? Consider...

Consider R with the cofinite topology. Is [0,1] compact? Can you describe the compact sets?

Consider R with the cocountable topology. Is [0, 1] compact? Can you describe the compact sets?

Consider R with the lower limit topology. Is [0,1] compact? Can you describe the compact sets?

Expert Solution

Colby Messinger answered 2 years ago

Topology (a) Prove that the interval [0,1] with the subspace topology is connected from basic principles....

Topology (a) Prove that the interval [0,1] with the subspace topology is connected from basic principles. (b) Prove that the interval [0,1] with the subspace topology is compact from basic principles.

Problem 17.5. Consider the function χ(0,1) : R → R (this is the characteristic of Definition...

Problem 17.5. Consider the function χ(0,1) : R → R (this is the characteristic of Definition 14.4). Find: (a) χ(0,1)((0,1)); (b) χ(0,1)((−1,3)); (c) and (in general) χ(0,1)((a,b)), where a,b ∈ R and a < b; prove that the set you found is correct; (d) χ−1 (0,1) ((−2,−1)); (e) χ−1 (0,1) ((0,2)); (f) and (in general) χ−1 (0,1) ((a,b)), where a,b ∈ R and a < b; prove that the set you found is correct, Definition 14.4 (for Problems 14.6 through...

Compact and analysis conception 1. Are all closed interval compact? for example [0,1]. are they closed...

Compact and analysis conception 1. Are all closed interval compact? for example [0,1]. are they closed and bounded? 2. If i can find the Maximum and Minimum, does that mean the set is closed and bounded?

Prove that the intersection of two compact sets is compact, using criterion (2).

Question: Prove that the intersection of two compact sets is compact, using criterion (2). Prove that the intersection of two compact sets is compact, using criterion (1). Prove that the intersection of two compact sets is compact, using criterion (3). Probably the most important new idea you'll encounter in real analysis is the concept of compactness. It's the compactness of [a, b] that makes a continuous function reach its maximum and that makes the Riemann in- tegral exist. For...

Calculate the relative (sub-space) topology with respect to the usual (metric) topology in R (the set...

Calculate the relative (sub-space) topology with respect to the usual (metric) topology in R (the set of real numbers), for the following sub-sets of R: X = Z, where Z represents the set of integers Y = {0} U {1 / n | n is an integer such that n> 0} Calculate (establish who are) the closed (relative) sets for the X and Y sub-spaces defined above. Is {0} open relative to X? Is {0} open relative to Y?

Let R be the real line with the Euclidean topology. (a) Prove that R has a...

Let R be the real line with the Euclidean topology. (a) Prove that R has a countable base for its topology. (b) Prove that every open cover of R has a countable subcover.

Prove the following Theorems: 1. A finite union of compact sets is compact. 2. Any intersection...

Prove the following Theorems: 1. A finite union of compact sets is compact. 2. Any intersection of compact set is compact. 3. A closed subset of a compact set is compact. 4. Every finite set in IRn is compact.

Let S ⊆ R be a nonempty compact set and p ∈ R. Prove that there...

Let S ⊆ R be a nonempty compact set and p ∈ R. Prove that there exists a point x_0 ∈ S which is “closest” to p. That is, prove that there exists x0 ∈ S such that |x_0 − p| is minimal.

Prove that a subspace of R is compact if and only if it is closed and bounded.

In RU (R is the reals, U is the usual topology), prove that any open interval...

In RU (R is the reals, U is the usual topology), prove that any open interval (a, b) is homeomorphic to the interval (0, 1). (Hint: construct a function f : (a, b) → (0, 1) for which f(a) = 0 and f(b) = 1. Show that your map is a homeomorphism by showing that it is a continuous bijection with a continuous inverse.)

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