Metric Spaces

Flashcards for pure module: Metric Spaces 2025-26

Axioms of a metric space

1. $ d(x,y) \geq 0 $

2. d(x,y)=0 implies x=y

3. d(x,y) = d(y,x)

4. $ d(x,z) \leg d(x,y) + d(y,z) $

5. \$x^2 + y^2 = z^2\$

Discrete Metric

$ d_{0} = \begin{cases} 1 &\text{if \(x \neq y)\} \\ 0 &\text{ if \(x = y)\ /end{cases}

Standard Metric on R

d_2(x,y) = |x-y|

Euclidian Metric

d_2(x,y) = sqrt((