let $F:M \to N$ be an injective immersion.Then,$F:M \to F(M)$ is a bijection and we can use it to give a topology on $F(M)$.Also,we know that every immersion is a local embedding.So,$F$ is a bijective map onto $F(M)$ which is also a local homeomorphism $\Rightarrow$F is a homeomorphism M onto $F(M)$.using this,homeomorphism,we can get differentiable atlas on $F(M)$ and hence $F(M)$ is called immersed submanifold.
My problem is:
"Suppose i give $F(M)$ subspace topology ,where,$F$ is injective immersion.Then,again $F$ becomes local homeomorphism+bijection onto $F(M)\Rightarrow$$F$ is a homeomorphism,but this is not so (counterexample:figure eight)".
where am i getting this wrong??
An injective immersion of a compact smooth manifold is an embedding. https://en.wikipedia.org/wiki/Immersion_(mathematics).
If the domain is non-compact then we need to add the condition that the injective immersion gives a homeomorphism $f: M \rightarrow f(M)$, where the topology on $f(M)$ is the sub-space topology inherited from $N$. In general, this will be different from the topology you described in the second sentence, and this is the source of confusion.
This is exactly the situation with the injective immersion $f:(0,1) \rightarrow \mathbb{R}^{2}$ parametrising the figure of eight figure. This is not a homeomorphism to its image (since for example the figure of eight is not simply connected), hence is not a smooth embedding. Note this is a bijective, local homeomorphism but it is not a homeomorphism.