The Earlier and the New Versions of the Definitions of Metric Spaces, Convergence, Continuity and Compactness Agree

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness.

(i) Metric spaces. For every set $X$ and every map $d:X\times X\to\mathbb{R}$, $d$ is a metric on $X$ in the earlier version if and only if it is a metric on $X$ in the new version; so $(X,d)$ is a metric space in the earlier version if and only if it is a metric space in the new one.

(ii) Separability. A metric space $(X,d)$ is separable in the earlier version if and only if it is separable in the new version, and a real normed space is separable in the earlier version if and only if it is separable in the new one.

(iii) Convergent sequences. For every metric space $(X,d)$, every sequence $(x_{m})$ in $X$ and every $x\in X$, $(x_{m})$ converges to $x$ in $(X,d)$ in the earlier version if and only if it converges to $x$ in the new version; hence it is convergent in the earlier version if and only if it is convergent in the new one.

(iv) Limits. For every metric space $(X,d)$ and every convergent sequence $(x_{m})$ in $(X,d)$, its limit in the earlier version is its limit in the new version.

(v) Open sets. For every metric space $(X,d)$ and every $U\subseteq X$, $U$ is open in $(X,d)$ in the earlier version if and only if it is open in $(X,d)$ in the new version.

(vi) Completeness. A metric space $(X,d)$ is complete in the earlier version if and only if it is complete in the new version.

(vii) Continuity. For all metric spaces $(X,d_{X})$ and $(Y,d_{Y})$, every $A\subseteq X$, every map $f:A\to Y$ and every $x\in A$, $f$ is continuous at $x$ relative to $A$ in the earlier version if and only if it is continuous at $x$ relative to $A$ in the new version, and $f$ is continuous on $A$ in the earlier version if and only if it is continuous on $A$ in the new one.

(viii) Compactness. For every topological space $(X,\mathcal{T})$ and every subset $A$ of $X$, $(X,\mathcal{T})$ is compact in the earlier version if and only if it is compact in the new version, and $A$ is compact in $(X,\mathcal{T})$ in the earlier version if and only if it is compact in $(X,\mathcal{T})$ in the new one.