A Short Introduction to Metric Spaces

Metric Space

Link: https://math.hws.edu/eck/metric-spaces/index.html#D-metric-space

Statements

Definition (Citation Needed). Let $X$ be a set and let $d:X\times X\to \mathbb{R}$ be a metric. The tuple $(X,d)$ is said to be a metric space.

Link to original

Open and Closed Sets

Link: https://math.hws.edu/eck/metric-spaces/open-and-closed-sets.html

Statements

Definition (@Tu11, pp. 317, @Eck23, Definition 1.1). Let $(X,d)$ be a metric space. For any $p\in X$ and $r>0$, the set $B_{<r}(p)=\{x\in X|m(x,p)<r\}$ is said to be an open ball with center $p$ and radius $r$.

Link to original

Statements

Definition. (@Eck23, Definition 1.2) Let $(X,d)$ be a metric space and let $x\in X$. A set $N\subset X$ is said to be a neighborhood of $x$ if there exists $\epsilon >0$ such that the open ball $B_{<\epsilon }(x)$ satisfies $B_{<\epsilon }(x)\subseteq N$.

Link to original

Statements

Definition. (@Eck23, Definition 1.1) Let $(X,d)$ be a metric space. A subset of $S$ is said to be open in $X$ if, for every $s\in S$, there exists $\epsilon >0$ such that the open ball $B_{<\epsilon }(x)$ satisfies $B_{<\epsilon }(s)\subseteq S$.

Link to original

Statements

Theorem (@Eck23, Theorem 1.2, @Men90, Theorem 6.4). Let $(X,d)$ be a metric space. Then, $(X,d)$ is a topological space, i.e., the metric open sets in $X$ satisfy the following properties:

• $X$ is open and $\varnothing$ is open.
• The union of any collection (finite or infinite) of open sets is open.
• The intersection of any finite collection of open sets is open.

Link to original

Statements

Definition. (@Eck23, Definition 1.4, @Tu11 pp. 319) Let be a $(X,d)$ be a metric space (resp. topological space). A set $S\subset X$ is said to be closed in $X$ if its complement $X\setminus S$ is open (resp. open) in $X$.

Link to original

Statements

Definition (@Eck23, Definition 1.5, @Men90, Defintion 6.6). Let $(X,d)$ be a metric space . A point $s\in S$ is said to be an accumulation point of $S$ if, for every $\epsilon >0$, the intersection of $S$ with the open ball $B_{<\epsilon }(s)$ is non-empty, i.e., $S\cap B_{<\epsilon }(s)\setminus \{s\})\ne \varnothing$.

Link to original

Statements

Definition (@Eck23, Theorem 1.4, @Men90, Definition 6.7). Let $(X,d)$ be a metric spaces. A set $S\subseteq X$ is closed if and only if every accumulation point of $S$ is an element of $S$.

Link to original

Statements

Definition (@Eck23, Definition 1.6). Let $(X,\tau )$ be a metric space, and $S\subseteq X$. The closure of $S$, denoted as $\mathrm{cl}(S)$, is the set of all elements $s\in S$ together with all the accumulation points of S.

Link to original

Statements

Theorem (@Eck23, Theorem 1.5). Let $(X,\tau )$ be a metric space, and $S\subseteq X$. The closure of $S$ is closed.

Link to original