concepts which is present in metric space but not in topological space I want to know some concepts which is present in metric space but not in topological space. The one that first comes to mind is uniform continuity, equicontinuity i.e. concepts defined with some kind of distance.  
 A: I'll give a somewhat unorthodox answer, deliberately interpreting the meaning of metric space in the broadest reasonable way possible. In Flagg's "quantales and continuity spaces" the concept of value quantale is introduced. A value quantale as an abstraction of the essential structure of the non-negative reals required to define a metric space. Then, given a value quantale $V$, a metric space is a set $X$ together with a distance function $d\colon X\times X\to V$ satisfying $d(x,x)=0$ and $d(x,z)\le d(x,y)+d(y,z)$, for all $x,y,z\in X$. 
In "A note on the metrization of spaces" (algebra universalis, to appear) I show that Flagg's metric spaces are equivalent to topological spaces in a precise sense (i.e., the category of all Flagg's metric spaces with continuous functions is equivalent to the category of topological spaces with continuous functions). That means that the classical definition of topological space and Flagg's definition of metric space are just different models of the same thing (i.e., of topology). 
So in particular, every topological space is metrizable for a suitable value quantale $V$. In light of that, concepts such as uniform continuity, completeness, etc. have meaning in topological spaces, but more often than not the interpretations of these concepts for the value quantale $V$ in the proof of the general metrization of a topological space become trivial.
In light of that, I will now interpret your question as follows. Which properties are special to metric spaces valued in the value quantale $[0,\infty ]$ of non-negative reals (such metric spaces are the usual ones (albeit not necessarily symmetric, nor separated)). To which my answer is that the concept of Lipschitz maps is undefined in a general value quantale, but is defined for $[0,\infty ]$ (but also in some other value quantales, so it's not special just to $[0,\infty ]$, but it is certainly far from generic). 
A: A very important difference between metric spaces and topological spaces is that metric spaces have large-scale structure while topological spaces have only local structure. (This is only true for the right interpretation of large-scale structure, in particular, one which doesn't include compactness.) This is the motivation for the very interesting field of coarse geometry, central to geometric group theory. 
The basic problem is that on any metric space $(X,d)$ the metric $d'=\min(d,1)$ defines the same topology as $d$. That's roughly why the open interval is topologically identical to the real line-even though they're metrically enormously different! Coarse geometry focuses on this large scale structure, so that now the real line becomes isomorphic to the integers, and every metric space defines a coarse space just as it defines a topological space. This is a nice way to make solid notions like the dimension of a lattice, which doesn't work at all topologically.
EDIT: I no longer endorse the claim that topological spaces have only local structure. Rather, they have no inherent notion of large scale structure, as a metric space achieves via its associated coarse structure. But topological spaces don't have a notion of small scale either, really-in metric spaces, such a notion arises from the associated uniform structure.
A: Cauchy sequences, complete space, sequential compactness as compactness, (no) real need of separation axioms less than $T_6$...
A: Without attempting to be exhaustive, in alphabetical order:


*

*balls

*boundedness and total boundedness

*Cauchy sequences/nets (make sense in topological vector spaces, too)

*completeness

*contractions

*Hausdorff distances

*isometry

*Lipschitz continuity

*uniform structures


Also, the following properties are often useful:


*

*Every metric space is perfectly normal. This implies the following separation axioms, too: normal, completely regular, regular, Hausdorff, and the singletons are closed ($T_1$).

*Every metric space is first countable. This implies, in particular, that closure and continuity can be dealt with in terms of sequences, as opposed to nets in general topological spaces.

*Every metric space is compact if and only if it is complete and totally bounded.

*Every compact metric space is separable.

*Every open set in a metric space is $F_{\sigma}$.

*Every closed set in a metric space is $G_{\delta}$.

*No complete metric space is a countable union of nowhere dense sets (Baire's category theorem). This property is actually topological in the sense it is preserved by homeomorphism.

*Separability and second countability are identical concepts for metric spaces.

*Also, compactness and sequential compactness are identical concepts.

A: Associated with a topological space $X$ is a  family of infinite cardinals,  that are invariant under homeomorphism, which are called topological cardinal functions. Some of these are:


*

*The weight $w(X),$ the least infinite cardinal $C$ such that $X$ has a base (basis) $B$ with cardinal $|B|\leq C.$

*The density $d(X),$ the least infinite cardinal $C$ such that $X$ has a dense subset $Y$ with $|Y|\leq C.$

*The cellularity, the least infinite cardinal $C$ such that $|F|\leq C$ for any pair-wise disjoint family $F$ of open subsets of $X.$

*The Lindelof Number $l(X),$ the least infinite cardinal $C$ such that any open cover $D$ of $X$ has a sub-cover $E$ with $|E|\leq C.$

*The tightness $t(X)$ (or $\tau(X)$), the least infinite cardinal $C$ such that whenever $p\in \overline A\subset X$ there exists $B\subset A$ with $|B|\leq C$ and $p\in \overline B.$

*The character $\chi(X),$ the least infinite cardinal $C$ such that every $p\in X$ has a local base (local basis) $B_p$  with $|B_p|\leq C.$

*For a $T_1$ space, the pseudo-character $\psi(X),$ the least infinite cardinal $C$ such that for each $p\in X$ there is a family $F_p$ of open sets with $|F_p|\leq C$ and $\{p\}=\cap F_p.$
When $f$ is a topological cardinal function, then $hf(X)$ (the "hereditary-$f$ of $X$") is $\sup \{f(Y):Y\subset X\}.$ We always have $hw(X)=w(X)\geq d(X)\geq c(X).$ 
For a metrizable space X we have $w(X)=d(X)=hd(X)=c(X)=hc(X)=l(X)=hl(X)$ and $\chi(X)=\psi (X)=\tau(X)=\aleph_0.$ Any or all of these inequalities may fail for a non-metrizable space.
If $\tau(X)\geq \aleph_1$ the closures of subsets of $X$ cannot be defined in terms of convergent countable sequences. 
For more about top'l cardinal functions, see General Topology by R.Engelking.
A: An interesting example is completeness. 
It's well known that the real numbers are complete. Completeness is the defining property of the reals, what makes the reals special. There are no missing elements. Every Cauchy sequence converges. 
The reals are topologically equivalent to the unit interval $(0,1)$ via the $\tan$ and $\arctan$ functions (suitably scaled). Yet $(0,1)$ is not complete. The Cauchy sequence $\{1/n\}_{n \in \mathbb{N}}$ fails to converge. 
Completeness is a metric property and not a topological one. 
