Editing Topological vector space (section)

===Non-vector topologies===

'''Discrete and cofinite topologies'''

If <math>X</math> is a non-trivial vector space (that is, of non-zero dimension) then the [[discrete topology]] on <math>X</math> (which is always [[Metrizable space|metrizable]]) is {{em|not}} a TVS topology because despite making addition and negation continuous (which makes it into a [[topological group]] under addition), it fails to make scalar multiplication continuous. The [[cofinite topology]] on <math>X</math> (where a subset is open if and only if its complement is finite) is also {{em|not}} a TVS topology on <math>X.</math>