Different uniformities giving the topology

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1. If X, Y , and Z are uniform spaces, f is uniformly continuous from X into Y , and g is uniformly continuous from Y into Z , prove that g ? f is uniformly continuous from X into Z.

2. Prove that any uniformity has a base consisting of symmetric sets.

3. For the real line with usual topology, find two different uniformities giving the topology. Hint: Give two metrics for the topology such that the identity is not uniformly continuous from one metric to the other.

Reference no: EM13990718

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