what is a usual topology for R i'm studying topology.
Let T be a usual topology for R(real) generated by usual metric.
first, i know that elements of T are open sets in R.
i wonder form of T(topology). 
Under given  condition, Can T have many form? or is T unique family?
 A: In general it is possible to place more than one topology on $\mathbb R$. For example, the following topology (the trivial topology) is a perfectly fine topology for $\mathbb R$:
$$
\{\varnothing,\mathbb R\}.
$$
(You should verify that it satisfies the axioms for a topology.) Here's another example, the discrete topology, which consists of all subsets of $\mathbb R$:
$$
2^\mathbb R.
$$
However, the topology you have in mind -- the standard topology on $\mathbb R$, which has as a base the open intervals $(a,b)$ -- is uniquely defined, although there is more than one way to describe it.
A: Expanding on what   Martín-Blas Pérez Pinilla said in a comment, “the usual metric” for $\Bbb R$ is the metric $$d(x,y) = \lvert x-y \rvert,$$ and “the usual topology” for $\Bbb R$ is the one where the basic open sets have the form $$(a,b) = \{ x\in \Bbb R\mid a < x < b \}.$$ (“Basic” here means that the open sets are unions of basic sets.)
There are many different possible topologies for $\Bbb R$, but only this one is called the “usual” topology.  There are many metrics one can put on $\Bbb R$, but only the $\lvert x-y\rvert $ metric is called the “usual” metric.
