So far I dealt with probability from a very intuitive point of view, like guessing frequencies etc. But while studying stochastic process (particularly with application to finance), I came across concepts like $\sigma$ algebra and a rigorous axiomatic foundation of probability.
The source I am reading from frequently refers to concepts of topology, Borel set, Hausdorff space etc. which sound kinda too advanced for me, since I am concerned mainly with the applications. Is there any way to understand stochastic calculus without going through topology? I have basic foundations of abstract, linear algebras, probability etc. Then can I get some simpler interpretation of axiomatic probability, martingale etc. without going through topological space?
I've studied quantitative finance, so I'll share my experience with you.
It's not essential to do a course in topology for financial modelling. In fact, most quants know very little about measure theory. Those who understand well usually work in academics. If you're interested to apply a financial model, it'd be better for you to focus more on:
I'm not saying measure theory is unimportant, it is important, but it shouldn't be your priority unless you want to be a mathematician. All financial theories built on measure theory (such as topology), but you can get away without understanding everything. For example, a standard text-book would model market information as filtration. While this is mathematically correct, you really don't need to do a course on it. You just need to know that it represents available information (e.g.: past stock prices) that can be used to price a financial instrument.
As you explore stochastic calculus, particularly with application to finance, you will find much more important stuffs to learn. I would focus my time on risk-neutral if I was you. You could always come back to topology later, but I don't need it to price instruments.