For a rigorous analysis course (think Baby Rudin), I believe that a good grasp of the concept of proof and set theory is required. The little bit of familiarity with topology (particularly that of the topology of $\mathbb{R}$) that you need will be covered concurrently in many analysis books. I've not used the book, but I hear How to Prove it is a good introduction to the transition from solving problems to proving theorems.

To move up to real analysis (Big Rudin, Royden, etc), the basics covered in Baby Rudin should be sufficient.

For topology, I find that Munkres is a great read and has many wonderful problems. Again, confidence with proof tactics is required.

To begin your analysis journey, perhaps a book like Abbott to introduce you to the basic topics and some basic proofs for typical analysis classes.

For a rigorous analysis course (think Baby Rudin), I believe that a good grasp of the concept of proof and set theory is required. The little bit of familiarity with topology (particularly that of the topology of $\mathbb{R}$) that you need will be covered concurrently in many analysis books. I've not used the book, but I hear How to Prove it is a good introduction to the transition from solving problems to proving theorems.

To move up to real analysis (Big Rudin, Royden, etc), the basics covered in Baby Rudin should be sufficient.

For topology, I find that Munkres is a great read and has many wonderful problems. Again, confidence with proof tactics is required.

These are essentially the books I used in classes progressing through analysis.

For a rigorous analysis course (think Baby Rudin), I believe that a good grasp of the concept of proof and set theory is required. The little bit of familiarity with topology (particularly that of the topology of $\mathbb{R}$) that you need will be covered concurrently in many analysis books. I've not used the book, but I hear How to Prove it is a good introduction to the transition from solving problems to proving theorems.

To move up to real analysis (Big Rudin, Royden, etc), the basics covered in Baby Rudin should be sufficient.

For a rigorous analysis course (think Baby Rudin), I believe that a good grasp of the concept of proof and set theory is required. The little bit of familiarity with topology (particularly that of the topology of $\mathbb{R}$) that you need will be covered concurrently in many analysis books. I've not used the book, but I hear How to Prove it is a good introduction to the transition from solving problems to proving theorems.

To move up to real analysis (Big Rudin, Royden, etc), the basics covered in Baby Rudin should be sufficient.

For topology, I find that Munkres is a great read and has many wonderful problems. Again, confidence with proof tactics is required.