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I need some notion about topology(I'm very interested in boundary points, open sets) and few examples of solved exercises about limits of functions($f:\mathbb{R}^{n}\rightarrow \mathbb{R}^m$) using $\epsilon, \delta$ and also some theory for continous functions. Please give me some links or name of the books which can help me.

Thanks :)

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Here is a fast free internet resource to get you started, while your text is in the mail.

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Topology by Klaus Janich is a good one as a general Topology textbook.

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If you don't want to go straight to general topology, you could look at a book more specifically about metric spaces, like Mícheál O'Searcoid's Metric Spaces.

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Well, most analysis books that I've seen (anything above high-school level maths) includes in the first and second chapter a discussion about $\mathbb{R}$ and the axioms, as well some notions about sets and point-set-topology, and then goes on to functions, limits, the $\epsilon - \delta$ criterion, and then in various order introduces differentiation and integration. Also, it seems that you're interested in $\mathbb{R}^n$ functions, so something about $\frac{\partial f}{\partial x}$ and Laplace transform, the $\nabla$ operator and so on.

You could try W. Rudin Elements of Real analysis and then perhaps find some text on Calculus II and III where they discuss vector and scalar fields, as well as higher-dimension derivatives and integrals. If you need a more heavy background in Topology than is provided by Rudin's Analysis chapters 1-6, than you can try Topology without Tears or some undergraduate text or course in Topology.

As for continuity of functions and the $\epsilon - \delta$ criterion, almost all calculus texts cover that, but if you want rigor, go for Real Analysis.

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$Topology$ by James Munkres is an excellent book for that sort of thing.

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