I am taking a class on Manifolds and whilst reading the book I came about some terminology that has got me confused. What is the differences/relations between these terms? Euclidian Space Topological Space Metric Space Vector Space I thought that a vector space was based on the premise that each coordinate described by its values are of an Euclidian Space, but the book I am reading acts as though they are completely two different things. Also, the other terms are used in Differential Geometry and I have no experience with them. So could you guys clarify, I would appreciate it a lot! :)
I suggest Wikipedia for the exact definitions. A metric space is a set equipped with a function that assigns a non-negative real number that you can think of as a "distance" to each pair of points in the set. This function is called a metric. For each x in X, and each r>0, the set [itex]B(x,r)=\{y\in X|d(x,y)<r\}[/itex] is called the open ball around x with radius r. A set that's equal to a union of open balls is said to be open. The most important thing the metric is used for is to define limits of sequences, but the definition can also be expressed in terms of open sets: Suppose that S is a sequence in X. A point x in X is said to be a limit of S, if every open set that contains x contains all but a finite number of terms of S. The metric is also used to define what it means to say that a function is continuous, but it turns out that continuity can also be defined entirely in terms of open sets: [itex]f:X\rightarrow Y[/itex] is said to be continuous if [itex]f^{-1}(E)[/itex] is open for each open [itex]E\subset Y[/itex]. These two observations provide the motivation for topological spaces. A topological space is a set together with a specfication of which of its subsets are to be called "open". (The specification must satisfy a short list of conditions). You can use that specification (the "topology") to define limits and continuity, without using a metric. Vector space...you probably know the definition already, so consider asking a more specific question. Euclidean space...There are several inequivalent definitions. I think what you need is the simplest one: n-dimensional Euclidean space is just the set [itex]\mathbb R^n[/itex] with the standard vector space structure.