  From studies of exotic quantum many-body phenomena to applications in
spintronics and quantum information processing, topological materials are
poised to revolutionize the condensed matter frontier and the landscape of
modern materials science. Accordingly, there is a broad effort to realize
topologically non-trivial electronic and photonic materials for fundamental
science as well as practical applications. In this work, we demonstrate the
first simultaneous site- and time- resolved measurements of a time reversal
invariant topological band-structure, which we realize in a radio frequency
(RF) photonic circuit. We control band-structure topology via local permutation
of a traveling wave capacitor-inductor network, increasing robustness by going
beyond the tight-binding limit. We observe a gapped density of states
consistent with a modified Hofstadter spectrum at a flux per plaquette of
$\phi=\pi/2$. In-situ probes of the band-gaps reveal spatially-localized
bulk-states and de-localized edge-states. Time-resolved measurements reveal
dynamical separation of localized edge-excitations into spin-polarized
currents. The RF circuit paradigm is naturally compatible with non-local
coupling schemes, allowing us to implement a M\"{o}bius strip topology
inaccessible in conventional systems. This room-temperature experiment
illuminates the origins of topology in band-structure, and when combined with
circuit quantum electrodynamics (QED) techniques, provides a direct path to
topologically-ordered quantum matter.
