Quartz sync: Mar 1, 2025, 2:26 PM

content/Definitions/Metric Spaces/Ball.md

@@ -0,0 +1,6 @@
+# Definition
+Let $d$ be a [[Metric|metric]] on a set $X$.
+
+The (open) **ball** with centre $x \in X$ and radius $r \geq 0$ is $B_{r} \equiv \{ y \in X | d(x,y) \gt r\}$.
+
+A sequence $\{ X_{n} \}$ in $X$ **converges** to $x \in X$ if it eventually belongs to any ball $B_{r}(x)$; $\forall r \gt 0 \; \exists N \in \mathbb{N}$ such that $\underbrace{d(x, x_{n})}_{x_{n} \in B_{r}(x)} \lt r, \; \forall n \gt N$.

content/Definitions/Metric Spaces/Interior Point.md

@@ -0,0 +1,2 @@
+# Definition
+Points such that small enough balls centred around them are contained in $A$.

content/Definitions/Metric Spaces/Metric Space.md

@@ -0,0 +1,6 @@
+# Definition
+A [[Metric]] on a set $X$ is a function $d : X \times X \to [ \, 0, \infty \rangle$ such that
+1. $d(x,y) = d(y, x), \; \forall x,y \in X$
+2. $d(x, y) = 0 \iff x = y$
+3. $d(x,z) \leq d(x,y) + d(y, z)$ (think of this as a triangle and Pythagoras' Theorem)
+Think of $d(x, y)$ as the distance between $x$ and $y$.