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Geometry

Topology

Analysis Warmup

         

Which of these is the set of all interior points of the open interval (0,1)?(0,1)?

Definition. An interior point of a set SS of real numbers is a number xx such that the open interval (xϵ,x+ϵ)(x - \epsilon, x + \epsilon) is contained in SS for some positive number ϵ.\epsilon.

Which of these is the set of all boundary points of the closed interval [0,1][0,1]?

Definition. A boundary point of a set SS of real numbers is a number xx such that every open interval (a,b)(a, b) contaning xx contains a number in SS and a number not in S.S.

Which of these is the set of all interior points of the closed interval [0,1]?[0,1]?

Definition. An interior point of a set SS of real numbers is a number xx such that the open interval (xϵ,x+ϵ)(x - \epsilon, x + \epsilon) is contained in SS for some positive number ϵ.\epsilon.

True or False?

The set of all boundary points (using the definition below) of the natural numbers N\mathbb{N} is N.\mathbb{N}.

Definition. A boundary point of a set SS of real numbers is a number xx such that every open interval (a,b)(a, b) containing xx contains a number in SS and a number not in S.S.

What is

n=3[1n,11n]?\bigcup_{n=3}^\infty \left[\frac{1}{n},1 - \frac{1}{n}\right]?

Notation. The bracket notation represents a closed interval, so [3,5]={x3x5}[3, 5] = \{x | 3 \leq x \leq 5\} is the set of all real numbers greater than or equal to 3 and less than or equal to 5.

The \bigcup symbol represents set union, so {1,2,3}{2,3,4}={1,2,3,4}.\{1, 2, 3\}\bigcup \{2,3,4\} = \{1, 2, 3, 4\}. The indexing n=3\bigcup_{n=3}^\infty represents an union of infinitely many sets, i.e.

n=3[1n,11n]=[13,113][14,114][15,115]\bigcup_{n=3}^\infty \left[\frac{1}{n},1 - \frac{1}{n}\right] = \left[\frac{1}{3},1-\frac{1}{3}\right] \bigcup \left[\frac{1}{4},1- \frac{1}{4}\right] \bigcup \left[\frac{1}{5},1- \frac{1}{5}\right] \bigcup \cdots

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