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Topology

Explore geometric properties and spatial relations that are unaffected by continuous deformations, like stretching and bending. The next time you scan a bar code on a can of soda, thank a topologist.

Topology of The Real Numbers

         

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

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

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

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

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

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

True or False?

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

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

What is

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

Notation. The bracket notation represents a closed interval, so \([3, 5] = \{x | 3 \leq x \leq 5\}\) 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 \(\{1, 2, 3\}\bigcup \{2,3,4\} = \{1, 2, 3, 4\}.\) The indexing \(\bigcup_{n=3}^\infty\) represents an union of infinitely many sets, i.e.

\[\bigcup_{n=3}^\infty \left[\frac{1}{n},1 - \frac{1}{n}\right] = \left[\frac{1}{3},1-\frac{1}{3}\right] \bigcup \left[\frac{1}{4},1- \frac{1}{4}\right] \bigcup \left[\frac{1}{5},1- \frac{1}{5}\right] \bigcup \cdots\]

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