- #1

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How can it be proved?

Thank's a lot,

Hedi

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- Thread starter hedipaldi
- Start date

- #1

- 210

- 0

How can it be proved?

Thank's a lot,

Hedi

- #2

- 22,129

- 3,298

- Let ##X## be a ##T_1## space (= singletons are closed). Let ##\{f_i~\vert~i\in I\}## be a collection of functions ##f_i:X\rightarrow X_i## which separates points from closed sets, then the evaluation map ##e:X\rightarrow \prod_{i\in I}X_i## is an embedding.

Now, your space ##X## has a countable basis consisting of clopen sets (why?). Use this to construct functions ##f_n:X\rightarrow \{0,1\}## for ##n\in \mathbb{N}## and apply the theorem.

- #3

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Thank you.

- #4

- 210

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The resulting function from X into the product space doesn't seem to be one-to-one.Maybe i fail to see something?

- #5

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After further thinking i suppose injectivity is due to X being Haussdorff.Am i right?

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