| Springer Online Reference Works |

| | The completeness of the metric **space** of continuous real-valued functions on a compactum is used, via the contraction-mapping principle, in the proof of the fundamental theorem on the existence of a solution to a differential equation under certain assumptions. |

| | The fact that duality between properties of a **topological** **space** and **topological** properties of the **space** of functions on them with the topology of pointwise convergence is inherited is of special significance. |

| | The quest for categories of **topological** **spaces** with nice categorical properties, such as Cartesian closedness, has also led to other ways of capturing the idea of "nearness" (in analysis, geometry and elsewhere) and has contributed to the emergence of categorical topology and the theory of **topological** structures, [a5]. |

| eom.springer.de /s/s086200.htm (966 words) |