Set

This note will be consistently updated. Related fields: Real Analysis, General Topology, Geometry.

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Supremum & Infimum

The supremum of a nonempty set $X \subset \mathbb{R}$ is the smallest scalar $y$ such that

\[y \geq x \text { for all } x \in X.\]

The infimum of a set $X \subset \mathbb{R}$ is the largest scalar $y$ such that

\[y \leq x \text { for all } x \in X.\]

If $\sup X \in X(\inf X \in X)$, then $\sup X=\max X(\inf X=\min X)$.

Example:

• $\sup \set{1 / n: n \geq 1}=\max \set{1 / n: n \geq 1}=1,$
• $\inf \set{1 / n: n \geq 1}=0 .$

1. Closed set
2. Bounded set
3. Compact set
4. Complete

For $\epsilon>0$ and $x \in \mathbb{R}^n$ we define $B_\epsilon(x)=\set{y \in \mathbb{R}^n:|x-y|<\epsilon}$ to be open ball with radius $\epsilon$ and center $x.$ Next, we collect further properties and terminologies for sets:

• A set $X \subset \mathbb{R}^n$ is called open if for every $x \in X$ there exists $\epsilon>0$ such that $B_\epsilon(x) \subset X.$
• A set $X \subset \mathbb{R}^n$ is closed if $\mathbb{R}^n \backslash X$ is open. Alternatively, we can define closedness of set as follows: For every sequence $\left(x^k\right)$ with $x^k \in X$ for all $k$ and $x^k \rightarrow x$, we have $x \in X.$
• A set $X \subset \mathbb{R}^n$ is bounded if there exists $B \in \mathbb{R}$ with $|x| \leq B$ for all $x \in X$.
• A bounded and closed set is called compact.

• The set ${(𝑥,𝑦)\mid 𝑥^2+𝑦^2<1}$ is bounded but not closed.
• The set ${(𝑥,𝑦)\mid 𝑥\ge0}$ is closed but not bounded.

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