© | Dror Bar-Natan: Classes: 2004-05: Math 1300Y - Topology: | (10) |
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this document in PDF: Agenda-040928.pdf

**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
.

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Dror Bar-Natan 2004-09-28