Is there a historical connection between $\sigma$-algebra and topology? Through learn about probability theory and topology, I feel that definition of $\sigma$-field(algebra) and topology are similar. From this, I also know about they are not same each other.
Furthermore, I interested about why they look like similar. Is there a historical connection between them? Is it result of convergent evolution? Or just by a chance?
I think that, if there are something (historical) reason for similarity, it will be strongly connected to how they different. So it will make me naturally understand about the difference.

These are different notions that can be used to capture different properties of functions (being continuous and measurable, respectively) but of course they are related, and one is often interested in settings that make this explicit.
Given any topological space $X$, we can create the least $\sigma$-algebra that contains the open sets of $X$ as members. This is called the Borel $\sigma$-algebra, and its members are called the Borel sets. As the name indicates, this notion was introduced by Émile Borel in 1896.
