But depending on the setup, often you just cant "sell the one thing at another exchange which you bought earlier at a different exchange", sure this depends on the country/etc.
There may be some odd financial instrument that can only be traded on a single exchange, but that's generally not going to be something that is liquid enough for HFT trading anyway. The idea that a stock is listed "on the NYSE" and can only be traded there is a quaint anachronism. e.g. https://help.tradestation.com/10_00/eng/tradestationhelp/rou...
