Is there a complete set of topological properties for a topological space? A property $P$ of topological spaces is called a topological property if for any two topological spaces $X$ and $Y$ which are homeomorphic, $X$ has property $P$ if and only if $Y$ has property $P$. Topological properties are useful because if one space has a certain topological property and another space does not, then the two spaces are not homeomorphic.
But I’d like to go in the other direction, namely showing that if two spaces have certain topological properties in common, then they are homeomorphic.  My question is, for a topological space $X$, does there exist a nontrivial collection of topological properties such that any space which has all these properties is homeomorphic to $X$?
By nontrivial I mean excluding properties like “homeomorphic to $X$”.

Every perfect, 2nd countable, zero dimensional, compact Hausdorff space is homeomorphic to the Cantor set.
$\{0,1\}^\mathbb{N}$ for example.  
Any order topology of a countable, order dense linear topology without end points is homeomorphic to the rationals.  The dyadic rationals for example.
All discrete spaces of cardinality $\kappa$ are homeomorphic.
Same for indiscrete spaces.  
All Hausdorff spaces of finite cardinality $n$ are homeomorphic.  
All not empty, discrete and connected spaces are homeomorphic.  
All spaces with exactly one open set are homeomorphic. 
