By Andrew Bernstein

ISBN-10: 0544179757

ISBN-13: 9780544179752

The unique CliffsNotes examine publications supply specialist statement on significant subject matters, plots, characters, literary units, and historic historical past. the most recent new release of titles during this sequence additionally good points glossaries and visible parts that supplement the vintage, normal format.

CliffsNotes on Atlas Shrugged is your consultant to writer Ayn Rand's masterpiece, an impassioned safeguard of the liberty of man's brain. She exhibits that with out the self sufficient brain, our society may cave in into primitive savagery.

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**Additional info for CliffsNotes on Rand's Atlas Shrugged**

**Example text**

The following notions play an important role in studying topological spaces. 48. Let X and Y be topological spaces. A function f : X → Y is called a homeomorphism from X onto Y if f is bijective, f is continuous on X , and the inverse function f −1 is continuous on Y . We say that X and Y are homeomorphic if there exists a homeomorphism from X onto Y . 49. Let a, b ∈ R be such that a < b. Then ]a, b[ and R are homeomorphic. Actually, it is easily verified that the function f : ]a, b[→ R defined by f (x) = 1 1 x−a + x−b for any x ∈ ]a, b[ is a homeomorphism from ]a, b[ onto R.

Prove the following. (a) lim inf n xn +lim inf n yn lim inf n (xn + yn ) lim supn (xn + yn ) lim supn xn + lim supn yn . lim inf n xn + lim supn yn whenever {xn + yn : n ∈ N } is (b) limn (xn + yn ) convergent. 94. Let (X, T ) be a topological space, and / X . Define T ∗ = {A ⊂ X ∗ : A ∈ T or X ∗ − A ∈ let X ∗ = X ∪ {δ}, where δ ∈ ∗ T and X − A is compact}. Prove the following. (a) T ∗ is a topology for X ∗ . (b) T is the relative topology on X induced by T ∗ . (c) (X ∗ , T ∗ ) is compact. 95. Let (X, T ) be a topological space, and let f : X → ] − ∞, ∞] be a function.

Let (X, T ) be a topological space and A ⊂ X . A cover A of A is said to be open if A⊂T. 14 (Lindelöf). Let (X, T ) be a topological space that has a countable base, and let A ⊂ X . Then any open cover of A contains a countable subcover of A. Proof. Let A be a open cover of A, and let B be a countable base for T . Since any set in A is a union of sets in B, there is a cover C ⊂ B of A such that each set in C is included in some set in A. For every C ∈ C, we choose DC ∈ A such that C ⊂ DC . Denote D = {DC : C ∈ C}.

### CliffsNotes on Rand's Atlas Shrugged by Andrew Bernstein

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