Equivalence Relations on [0,1]x[0,1] and Hausdorff Spaces

[lttlbbygurl]

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]

Wait, so (0,0) is not equivalent to itself? Then it's not an equivalence relation?

 

[Martin Rattigan]

Wait, so (0,0) is not equivalent to itself? Then it's not an equivalence relation?

I think topologists usually don't bother explicitly describing eqivalence classes with a single member, so in this case each {(0,y)} would be a singleton equivalence class. (What can you say about open sets around {(0,y)} and {(0,y')} in the quotient topology, where y and y' are distinct?)