add in product topology basis proof

topology/product_topology.html

@@ -63,12 +63,18 @@ <h1>product topology</h1>
             <div class="proof">
                 <div class="title"></div>
                 <div class="content">
-                    <p>
-                        To show it's a basis we will use the <span class="knowledge-link" data-href="/topology/topological_spaces.html#theorem-basis-criterion">basis criterion</span>. So let `U` be an open set of `X xx Y`,
-                    </p>
+		    <details>
+			<summary>show</summary>
+			    <p>
+				To show it's a basis we will use the <span class="knowledge-link" data-href="/topology/topological_spaces.html#theorem-basis-criterion">basis criterion</span>. So let `W` be an open set of `X xx Y`, and let `(x, y) in W`, since the product topology is generated by the basis `{U xx V: X in cc T_X text( and ) Y in cc T_Y}`, then we know that by the definition of a topology generated by a basis that there is an element `U xx V` such that `(x, y) in U xx V sube W`.
+			    </p>
+			    <p>
+			    Since we assumed that `cc B` and `cc C` were <span class="knowledge-link" data-href="/topology/topological_spaces.html#definition-basis-for-a-topology">bases for</span> `X` and `Y` respectively then, we know that there is some element `B in cc B` such that `x in B sube U` and there is some `C in cc C` such that `y in C sube V`, thus we have found an element `B xx C in cc D` such that `(x, y) in B xx C sube U xx V = W` which proves that `cc D` is a basis and that it generates the toplogy of `X xx Y`
+			    </p>
+		    </details>
                 </div>
             </div>
 
         </div>
     </body>
-</html>
+</html>