Time-variance and causality of $y(t)=\sum_{k=-\infty}^{+\infty}x(t)\delta(t-kT)$ Is the following system time invariant and/or causal?
  
  $$y(t)=\sum_{k=-\infty}^{+\infty}x(t)\delta(t-kT)$$


I think that it isn't time invariant and it is causal, and my thought is based on the fact that this is actually a sampling system that samples on $T-$intervals the input function. Is this correct and if yes, how can the time-variance be formally proven?
I agree with you on the system being a causal system since at a point in time $t$, the output is only made out of the input signal at the same point in time, weighted with an impulse (or not).

You can prove its time-variance by showing that a time-shifted input signal yields another output signal than the original input signal, for example:


$x(t) = \delta(t-\frac{T}{2})$ evaluates to $y(t)=0$ while
$x(t) = \delta(t)$ does not.