How can one show that the paths of the standard Wiener process are continuous in $T$ with probability one? Can we just proof it with the assumption of independence ? Thank You in advance!
Using the distribution and independence of increments allows to prove $L^2$ (mean-square) continuity. Proving the a.s. continuity is much harder. Paul Lévy's construction of Brownian motion is related in Le Gall; an alternative is to construct the Brownian motion through Haar wavelet functions or Fourier series.
It's part of the definition
I'd just like to re-iterate the comment by Kevin, (which as far as I can tell is the answer). There are three properties which define a standard Brownian motion / Wiener process:
- Independent increments.
- Normally distributed with variance equal to the time increment.
- The path is continuous.
Which hopefully any "standard" textbook on stochastics will re-iterate (Klebaner, Kloeden and Platen, Shreve, Oksendal, etc.).
However, as remarked in this comment, it is possible to drop this assumption and start with alternative constructions/definitions, from which continuity might be a consequence rather than a postulate. However, I suspect this is both more advanced, more nuanced, and less standard, so I don't know any references for this starting point.