Order of a study

Firstly I would like to clear up some things that I disagree with that have been mentioned in the comments.

To the contrary to some things mentioned above (in the comments), I think that taking topology before real analysis can be incredibly advantageous, since real analysis relies heavily on topological concepts (as do many areas of math).

As an undergraduate, I took my first analysis class before topology. I wouldn't recommend this, many of the tools involved were above my head, and many of the arguments used in analysis are topological in nature. I took topology before entering my second analysis class and the difference was amazing. Also a typical introductory course in topology does not really build off of the language of algebraic structures... it should be self contained (this is in response to a comment made). Although being familiar with abstract mathematical structures is advantageous, it shouldn't be necessary at all. Topology is, after all, one of the main building blocks of mathematics.

Now , in response to your inquiry:

Most introductory math courses are self contained in that they usually at least briefly discuss methods of proofs with some examples. However, having a firm grasp on logic will make any proof-based mathematics class a lot simpler. We have two cases:

Case 1: Topology before logic

I am trying to think of any benefits for this case. The only situation where I can see this as beneficial is if your math logic class were to draw examples from topology. For example, after discussing proof by contradiction maybe something like the following example could arise:

Prove by contradiction that there is no smooth injective map