Quotient space (topology): Difference between revisions

The '''quotient space''' under <math>\,\sim\,</math> is the quotient set <math>Y</math> equipped with the '''quotient topology''', which is the topology whose [[open set]]s are the all those [[subset]]s <math>U \subseteq Y = X / {\sim}</math> such that <math>\{ x \in X : [x] \in U \} = \cup_{u \in U} u</math> is an [[Open set|open subset]] of <math>\left(X, \tau_X\right);</math> that is, <math>U \subseteq X / {\sim}</math> is open in the quotient topology on <math>X / {\sim}</math> if and only if <math>\{ x \in X : [x] \in U \} \in \tau_X.</math>