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Comment. is continuous at iff for every neighborhood of , its inverse image contains a neighborhood of .
Agenda. We will discuss two primary notions and the interaction between them and along the way also learn about sequences....
First notion -- the product topology. (The naive definition and the box topology), definition by listing our requirements, uniqueness and existence, interaction with the trivial topology, the subspace topology, and the discrete topology.
Second notion -- metric spaces and metrizability Definition, examples, the metric topology, -ness, metrizability.
The interaction We'll prove three theorems:
Theorem 1. (good) is metrizable iff every is metrizable.
Theorem 2. (who cares?) is not metrizable.
Theorem 3. (bad) is not metrizable.
In order to prove Theorems 2 and 3 we will need to know about sequences, and these are quite interested by themselves:
Sequences. Convergence, sequential closure.
Proposition 1. The sequential closure is always a subset of the closure, and in a metrizable space, they are equal.
Proposition 2. If and is metric, then is continuous iff for every sequence in , the convergence implies the convergence .