If $\{N(t)\}_{t \geq 0}$ is a Poisson process with rate $\lambda$, then the number of arrivals in $[0, t]$, called $N(t)$, is a Poisson random variable with parameter $\lambda t$. Further, the inter-arrival times are exponential random variables with mean $ = \frac{1}{\lambda}$.
I read that given the event that there is exactly one arrival in $[0, t]$, the arrival time $T$ is distributed uniformly on $[0, t]$ - conditioned on the event.
Further, given the event that there are $k$ arrivals in $[0, t]$, the arrival times are distributed independently and uniformly in the interval $[0, t]$.
Suppose the arrival times are $T_1$, $T_2$, ..., $T_k$. How are they independent? For example, if I know $T_1 = \lambda$, I definitely know that $T_2$ must take values in $[\lambda, t]$. So $T_1$ and $T_2$ are not independent.
What is wrong with this interpretation?