Topological Property and Algebraic Property What do you mean by topological property ? Similarly , by algebraic property , what do we mean ? Please clarify the difference between them. 
 A: It usually means properties that are preserved under isomorphisms of topological spaces (homeomorphisms) and isomorphisms of algebras respectively.
A: A set X can be given a topology (topological structure) by specifying some collection of subsets to be the open sets; if the elements of X have some natural notions of similarity or distance between them, the topology can describe that. Similarly, it can be given some algebraic structure by means of some functions like $m: X \times X \to X$ (or $X^n \to X$ more generally) that satisfies certain properties (e.g. to be a group, commutative ring, $k$-algebra); this happens if there are natural ways to add or multiply the objects of $X$. 
In the settings where there is a natural concept of both, there are often some compatibilities (i.e. the multiplication $m$ can be continuous).
Example:
The real numbers have a topology and the structure of a field. These two structures are compatible, since multiplication, inversion and addition are continuous functions. There can be transformations of $R$ that preserve something of the topology (paths in $R^n$), but not necessarily the additive structure of $R$ (if you are thinking of $R^n$ with its natural vector space structure, then the path must be a line) or the field structure (unless you are embedding $\mathbb{R}$ into the complex numbers $\mathbb{C}$)). Sometimes something of both are preserved, an example of this is a flow along a (nice) vector field.
Another example are matrix groups (Lie groups). For example, the set of all invertible $nxn$ real matrices has both a topological structure (since it is naturally a subset of some big Euclidean space, $R^{n \times n})$, and an algebraic structure, because you can multiply and invert such matrices.
