I.1.7c

12/24/20

Merry fuck you eve. This one is quick

Let Y be a subspace of X, where X is Noetherian. We want to show that Y is Noetherian.

Suppose we have a chain

| (1) |

of closed subsets of Y

Note, by the subspace topology, for each i = 1,2,…:

| (2) |

for some C_{i} closed in X.

For each i, let

| (3) |

and note that each D_{i} is closed in X.

Then the chain

| (4) |

stabilizes since X is Noetherian. So for some n,

| (5) |

Note: ∀k < i, Y _{k} ⊃ Y _{i}, so

Y _{i} | = Y _{i} ∩ Y _{i-1} ∩∩ Y _{1} | (6) | ||

= (Y ∩ C_{i}) ∩ (Y ∩ C_{i-1}) ∩∩ (Y ∩ C_{1}) | (By (2)) | (7) | ||

= Y ∩ C_{i} ∩ C_{i-1} ∩∩ C_{1} | (8) | |||

= Y ∩ D_{i} | (9) |

But intersecting Y with (5) yields

Y ∩ D_{n} = Y ∩ D_{n+1} = | (10) | |

Y _{n} = Y _{n+1} = | (using (9)) | (11) |

(12) |

So we're done. WELP, THE EQUATION NUMBERS ARE ALL FORMATTED FUCKED UP, BUT FUCK IT.