I already know the base definition and 4 axioms. The things I don't know are completeness, compactness, and connectivity.Also, in the example:$\gamma: [0, 1]\to D$, $\gamma(0) = z_1, \gamma(1) = z_2$ (where $D$ is metric space, and $z_1$ and $z_2$ are points),"density metric in D" $\varphi(z)$ is mentioned,
$\varphi : D\to [0, +\infty)$,
$\varphi(\gamma) = \int_\gamma {\varphi(z)\left|dz \right|}$
also mentioned "geodesic line" $\varphi = \int_a^b{\sqrt{1+(f'(x))^2}dx}$
The limit value of strings in metric space seems intuitive. The terms completeness and compactness are familiar to me from other fields, but the density of metrics, the appearance of integrals and the geodesic line completely confuse me.
I couldn't find any material where these are concisely and well explained. Can anyone explain what are these? Course I am having this in is Real Analysis I, and although metric spaces are done more in detail on Real Analysis II, I guess this should just be an introductory level but it still doesn't seem trivial to me except the basic definition of a metric and metric space.