I am a little confused about the properties (is it Linear, Causal, Time-Invariant, Stable?) of this T system.
$$T[x(t)]=\sum_{k=t_0}^{t}{x(k)}$$
Some are obvious (it is linear), but can’t come up with the rest of them… any tips on this would be appreciated.
The system can't be time-invariant, because the interval over which the input is accumulated depends on the time $t$.
The formula implies that the system's output is zero if $t<t_0$. If this is the case then the system is causal, because no future values of the input signal are necessary for computing the output signal.
As for stability, the system is definitely BIBO stable because a bounded input will always produce a bounded output.