I'm probably in a very slim minority of people who took their first topology course before they took their first real analysis course, and I think my understand

I'm probably in a very slim minority of people who took their first topology course before they took their first real analysis course, and I think my understanding of both topics improved as a result of taking them in this order. there's no way to actually avoid epsilon delta arguments in real analysis, but it's helpful to know that there is a more "intuitive" way of thinking about continuity (although admittedly it's a little weird when you first encounter it), that requires a lot less algebraic magic. Anyway, there's more to real analysis than the topology of the real numbers, but I think it's a great starting place.