- State the definition of the product topology, the subspace topology, and the quotient topology.
Solution
1. The product topology on \(X\times Y\) is the topology with basis \(\{U\times V:U\subseteq X\text{ open},\ V\subseteq Y\text{ open}\}\).
This is the defining basis for the product topology.
2. If \(A\subseteq X\), the subspace topology on \(A\) is \(\{A\cap U:U\subseteq X\text{ open}\}\).
This is the definition of the topology induced by the inclusion \(A\hookrightarrow X\).
3. If \(q:X\twoheadrightarrow Y\) is a surjection, the quotient topology on \(Y\) is defined by\[U\subseteq Y\text{ open}\iff q^{-1}(U)\subseteq X\text{ open}.\]
This is the definition of the quotient topology induced by \(q\).