Maxwell’s demon and the physical nature of space

Maxwell’s demon is a thought experiment to examine the second law of thermodynamics. But is not only about a special part of physics, but about some unspoken features of our Universe as well.
It is a small, intelligent entity who can detect the speed of individual molecules and opening or closing a door that divides a box into two parts, he/she is able to separate the fast molecules into one while the slow ones into the other part of the box. The result is the decrease of entropy in a closed system – that is contradicts the second law.
Or not, since, according to an argument based on information theory, the main problem is that if Maxwell’s demon’s memory isn’t infinitely large, then sooner or later information should be erased form it.This process emits heat into the box, because information erasing necessarily causes heat [Charles Seife: Decoding the Universe, p. 85.]. So the second law remains valid, since the entropy grows in the closed system.
This answer is partly based on the presupposition that only finitely large memories are possible and includes another presupposition, too, about the nature of space–after all, if you would be able to divide the space into infinitely small amounts, then it would be possible to store an infinitely large amount of information in a finite storage (at least, theoretically). I.e. If you have a two square cm surface of data storage, then you could use the first square cm to store the first piece of data; a half square cm to store the second piece of data; etc. ad infinitum.
An infinitely large memory perhaps not as unreal as it seems to be for the first sight, since there are plausible theories about hypercomputing based on relativistic spacetime of blackhole physics (see the details here). In this case manipulating an infinitely huge amount of data in an finite period of time is possible thanks to the nature: since it is presupposed that time is divisible infinitely many pieces, the result is that we have enough time to perform infinitely many operations in a finite period of time.
Ad analogiam: We can hypothesize that the space’s nature is similar and it is divisible infinitely, so it is possible to build a spatially finite storage to store an infinite amount of information. So we never should erase a bit of information–and the entropy wouldn’t rise in the box.
Obviously, nobody knows whether the space continuous, but it has been shown by this example how the laws of thermodynamics is embedded into the "environment" of the existing physical laws. And we cannot exclude the existence of Maxwell’s demon if the space we live in is not discrete, but continuous.