Question about euclidian topology I have 2 questions about euclidian topology.
1.(The silly one) Why it is termed euclidian?
2. Why open intervals are used in the definition of euclidean  topology rather than closed intervals? Because I think they could serve(?) the same purpose in euclidean topology.
 A: To answer your first question, Euclidean distance is a geometric notion that should be familiar, and that notion dates back to Euclid, who pioneered what we call "Euclidean Geometry". The relation between Euclidean Geometry and the Euclidean topology is somewhat subtle. It is in fact the topology generated by the one-dimensional "Euclidean Metric". Metric spaces were invented as ways to talk about "distance" between objects. The Euclidean Metric on $\mathbb{R}^n$, for any $n \geq 1$ uses the notion of "distance" in $n$-dimensions, determined by the Pythagorean Theorem for two and three dimensions. So in one dimension, if you have a point $b$ and a point $a$, the distance between them is determined by $|b-a|$. Open intervals essentially function as the "one dimensional open balls" of this metric space.
To answer the second question, Topology was greatly motivated by generalizing what is called "The Euclidean Topology" A general notion of being "open" was inspired by an "open" interval and "closed" was inspired by "closed intervals". So that more or less indicates how closed intervals do not serve the same function.
Part of what I feel is vital to understanding general topology is understanding the roll that "limit points" play in generalizing the notion of epsilon-delta limits in analysis. Limit points are supposed to convey the structure of what happens when you zoom in arbitrarily close on a point using open sets as your binoculars, so-to-speak.
A closed set contains all its limit points just as a closed interval contains all its endpoints, and everything in between. The Euclidean Topology in principle conveys that message, since that is how the real line is supposed to work.
If this philisophical answer doesn't satisfy your curiosity, there's a mathematical reason why closed intervals don't serve the same function. The topology $\mathcal{T}_C$ on $\mathbb{R}$ generated by the basis of closed intervals is NOT the Euclidean topology on $\mathbb{R}$. (as Jack Davies pointed out). Note that closed intervals are not open in the Euclidean topology, but they are in $\mathcal{T}_C$. Also note that open intervals are also open since they are a union of closed intervals.
This topology $\mathcal{T}_C$ is different by definition, and it's not just different. It's very different. $\mathcal{T}_C$ is in fact just $\mathcal{P}(\mathbb{R})$ (the power set topology on any set is called the "discrete topology") since the singleton sets are closed intervals, and the singleton sets plus the empty set form a basis on this topology. And if you look at the limit points of a given open set, you will see that NOTHING is a limit point, since singleton sets contain nothing else other than the point, so your binoculars verify no point as a limit point. Analysis now becomes very boring. We want analysis to be fun so we use the Euclidean topology.
