# Summary of Topology on P-adic Numbers

## Theorem

Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Let $\tau_p$ be the topology induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$.

Then $\struct{\Q_p, \tau_p}$ is:

$(1): \quad$ Hausdorff
$(2): \quad$ second-countable
$(3): \quad$ totally disconnected
$(4): \quad$ locally compact

## Proof

Follows from:

P-adic Numbers is Hausdorff Topological Space
P-adic Numbers is Second Countable Topological Space
P-adic Numbers is Totally Disconnected Topological Space
P-adic Numbers is Locally Compact Topological Space

$\blacksquare$