We will examine three threads of inquiry in topology: convergence/compactness properties, spaces built out of other spaces (i.e. the space of real-valued continuous functions or the hyperspace of closed sets), and topological games. When a space is built out of another space, we can often translate the topological information from the first space to the second. For instance, open covers of the space can produce clustering sequences of real-valued functions. This topological information can be encoded through strategies in certain topological games. Working with Chris Caruvana and Steven Clontz, we have developed techniques for tying all of these threads together and have proven a wide array of connections between spaces and common constructions on those spaces. The general theory will be discussed and specific examples will be displayed.