Metric Topology

Definition

Metric Topology

A metric topology is a Topology generated by a basis defined by a Metric $d$. ${\mathcal{B}}_d=\{B_d(x,ϵ)|x\in X,ϵ>0\}$ where $B_d(x,ϵ)=\{y\in X|d(x,y)<ϵ\}$ is an $ϵ$-ball centered at $x$.

Metrizable Topology

A Topological Space $(X,\mathcal{T})$ is metrizable If there exist a Metric $d$ on $X$ that induces the topology $\mathcal{T}$.

Facts

Consider two metric topologies $(X,\mathcal{T},d),(X,{\mathcal{T}}^{\prime },d^{\prime })$. $\mathcal{T}\subset {\mathcal{T}}^{\prime }⟺\forall x\in X,ϵ>0,\exists \delta >0\text{s.t.}{B}_{d^{\prime }}(x,\delta )\subset B_d(x,ϵ)$

A metrizable Topological Space $(X,\mathcal{T})$ is Normal Space

Every Metric Space is a first-countable space.

Link to original

A metric space $(X,\mathcal{T})$ is Compact if and only if the space has Bolzano–Weierstrass Property

A metric space $(X,\mathcal{T})$ is Compact if and only if the space is complete and totally bounded.

A Metric Space is a completely normal space

Link to original