- #1
lttlbbygurl
- 6
- 0
We have a equivalence relation on [0,1] × [0,1] by letting (x_0, y_0) ~ (x_1, y_1) if and only if x_0 = x_1 > 0... then how do we show that X\ ~is not a Hausdorff space ?
g_edgar said:Wait, so (0,0) is not equivalent to itself? Then it's not an equivalence relation?
An equivalence relation on [0,1]x[0,1] is a relation that satisfies three properties: reflexivity, symmetry, and transitivity. This means that for any points (a,b), (c,d), and (e,f) in [0,1]x[0,1], the relation must have the following properties: (a,b) is related to (a,b), (a,b) is related to (c,d) if and only if (c,d) is related to (a,b), and if (a,b) is related to (c,d) and (c,d) is related to (e,f), then (a,b) is related to (e,f).
Hausdorff spaces are topological spaces in which any two distinct points can be separated by disjoint open sets. In other words, for any two points (a,b) and (c,d) in the space, there exist open sets U and V such that (a,b) is in U, (c,d) is in V, and U and V do not intersect. Equivalence relations on [0,1]x[0,1] can be used to define a topology on the space, and if this topology satisfies the Hausdorff condition, then the space is a Hausdorff space.
Yes, an equivalence relation can be used to partition [0,1]x[0,1]. This means that the space can be divided into disjoint subsets, called equivalence classes, where each point in a subset is related to every other point in the subset. In other words, the relation creates a grouping of points in the space based on their equivalence.
The quotient space is a space that is created by identifying points in [0,1]x[0,1] that are related by an equivalence relation. This means that the quotient space is formed by collapsing each equivalence class into a single point. The resulting space has the same topological properties as the original space, but with fewer points.
Yes, equivalence relations on [0,1]x[0,1] and Hausdorff spaces have many applications in mathematics and other fields. They are commonly used in topology, geometry, and group theory, among others. In particular, the concept of equivalence relations is fundamental in understanding and defining group actions, which have a wide range of applications in physics, computer science, and other areas.