NaOH by electrolysis

So, when water is electrolysed NaOH is created,but since NaOH is an electrolyte itself how de we know that the NaOH doesn't start electrolysing too?

 

Sodium chlorate and some other stuff

 

I thought somebody had experience and could tell me this from experience, I tried making it a lot of time and it allways gets contaminated from the collapsing of my carbon electrodes. So before i find out if will stop when its done (so i know when its done) I wont go and find a power supply that doesnt collapse my electrodes,also i need to make a cell membrane

 

Allready did,I'm just going to use it for getting a small amount of pure sodium