From 334514fd2ee644794087cbe144bef0f7d324969e Mon Sep 17 00:00:00 2001 From: yhx-12243 Date: Fri, 6 Dec 2024 08:13:06 +0800 Subject: [PATCH] Noetherian spaces (cont'd) (#1014) --- spaces/S000015/properties/P000016.md | 12 ------------ spaces/S000015/properties/P000130.md | 10 ---------- spaces/S000015/properties/P000208.md | 7 +++++++ spaces/S000016/properties/P000016.md | 11 ----------- spaces/S000016/properties/P000130.md | 10 ---------- spaces/S000016/properties/P000208.md | 7 +++++++ spaces/S000019/properties/P000208.md | 7 +++++++ spaces/S000045/properties/P000208.md | 7 +++++++ spaces/S000048/properties/P000016.md | 12 ------------ spaces/S000048/properties/P000208.md | 7 +++++++ spaces/S000150/properties/P000208.md | 7 +++++++ spaces/S000151/properties/P000208.md | 7 +++++++ spaces/S000200/properties/P000016.md | 7 ------- spaces/S000200/properties/P000208.md | 7 +++++++ theorems/T000251.md | 7 +++++-- theorems/T000651.md | 9 +++++++++ theorems/T000658.md | 11 +++++++++++ theorems/T000659.md | 13 +++++++++++++ theorems/T000660.md | 13 +++++++++++++ 19 files changed, 107 insertions(+), 64 deletions(-) delete mode 100644 spaces/S000015/properties/P000016.md delete mode 100644 spaces/S000015/properties/P000130.md create mode 100644 spaces/S000015/properties/P000208.md delete mode 100644 spaces/S000016/properties/P000016.md delete mode 100644 spaces/S000016/properties/P000130.md create mode 100644 spaces/S000016/properties/P000208.md create mode 100644 spaces/S000019/properties/P000208.md create mode 100644 spaces/S000045/properties/P000208.md delete mode 100644 spaces/S000048/properties/P000016.md create mode 100644 spaces/S000048/properties/P000208.md create mode 100644 spaces/S000150/properties/P000208.md create mode 100644 spaces/S000151/properties/P000208.md delete mode 100644 spaces/S000200/properties/P000016.md create mode 100644 spaces/S000200/properties/P000208.md create mode 100644 theorems/T000651.md create mode 100644 theorems/T000658.md create mode 100644 theorems/T000659.md create mode 100644 theorems/T000660.md diff --git a/spaces/S000015/properties/P000016.md b/spaces/S000015/properties/P000016.md deleted file mode 100644 index d797637ad6..0000000000 --- a/spaces/S000015/properties/P000016.md +++ /dev/null @@ -1,12 +0,0 @@ ---- -space: S000015 -property: P000016 -value: true -refs: -- doi: 10.1007/978-1-4612-6290-9_6 - name: Counterexamples in Topology ---- - -If $\mathcal{U}$ is an open cover an $U \in \mathcal{U}$, then $X \setminus U$ is finite, and can be covered by finitely other members of $\mathcal{U}$. - -See item #2 for space #18 in {{doi:10.1007/978-1-4612-6290-9_6}}. diff --git a/spaces/S000015/properties/P000130.md b/spaces/S000015/properties/P000130.md deleted file mode 100644 index 4b39296959..0000000000 --- a/spaces/S000015/properties/P000130.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000015 -property: P000130 -value: true -refs: -- doi: 10.1007/978-1-4612-6290-9_6 - name: Counterexamples in Topology ---- - -As stated in item #2 for space #18 in {{doi:10.1007/978-1-4612-6290-9_6}} every subset of $X$ is compact. Hence every neighborhood of every point is compact and every point has a local base of compact neighborhoods. diff --git a/spaces/S000015/properties/P000208.md b/spaces/S000015/properties/P000208.md new file mode 100644 index 0000000000..79a2febdeb --- /dev/null +++ b/spaces/S000015/properties/P000208.md @@ -0,0 +1,7 @@ +--- +space: S000015 +property: P000208 +value: true +--- + +The closed sets satisfy the descending chain condition, because all closed sets are finite except $X$. diff --git a/spaces/S000016/properties/P000016.md b/spaces/S000016/properties/P000016.md deleted file mode 100644 index 49607cb6bb..0000000000 --- a/spaces/S000016/properties/P000016.md +++ /dev/null @@ -1,11 +0,0 @@ ---- -space: S000016 -property: P000016 -value: true -refs: -- doi: 10.1007/978-1-4612-6290-9_6 - name: Counterexamples in Topology ---- - - -See item #7 for space #19 in {{doi:10.1007/978-1-4612-6290-9_6}}. diff --git a/spaces/S000016/properties/P000130.md b/spaces/S000016/properties/P000130.md deleted file mode 100644 index 4101e3f645..0000000000 --- a/spaces/S000016/properties/P000130.md +++ /dev/null @@ -1,10 +0,0 @@ ---- -space: S000016 -property: P000130 -value: true -refs: -- doi: 10.1007/978-1-4612-6290-9_6 - name: Counterexamples in Topology ---- - -As stated in item #2 for space #19 in {{doi:10.1007/978-1-4612-6290-9_6}} every subset of $X$ is compact. Hence every neighborhood of every point is compact and every point has a local base of compact neighborhoods. diff --git a/spaces/S000016/properties/P000208.md b/spaces/S000016/properties/P000208.md new file mode 100644 index 0000000000..05b0d4fd42 --- /dev/null +++ b/spaces/S000016/properties/P000208.md @@ -0,0 +1,7 @@ +--- +space: S000016 +property: P000208 +value: true +--- + +The closed sets satisfy the descending chain condition, because all closed sets are finite except $X$. diff --git a/spaces/S000019/properties/P000208.md b/spaces/S000019/properties/P000208.md new file mode 100644 index 0000000000..88bde9c296 --- /dev/null +++ b/spaces/S000019/properties/P000208.md @@ -0,0 +1,7 @@ +--- +space: S000019 +property: P000208 +value: false +--- + +The closed sets different from $X$ are exactly the compact sets in {S25}. However, they don't satisfy the descending chain condition: we have $Y_1 \supsetneq Y_2 \supsetneq \cdots$ where $Y_n = \left\{ 0, \frac 1n, \frac 1 {n + 1}, \frac 1 {n + 2}, \dots \right\}$. diff --git a/spaces/S000045/properties/P000208.md b/spaces/S000045/properties/P000208.md new file mode 100644 index 0000000000..6f63c5ebc5 --- /dev/null +++ b/spaces/S000045/properties/P000208.md @@ -0,0 +1,7 @@ +--- +space: S000045 +property: P000208 +value: false +--- + +$\left[ -1, \frac n {n + 1} \right)$ is a strictly increasing sequence of open sets. diff --git a/spaces/S000048/properties/P000016.md b/spaces/S000048/properties/P000016.md deleted file mode 100644 index 51528a89da..0000000000 --- a/spaces/S000048/properties/P000016.md +++ /dev/null @@ -1,12 +0,0 @@ ---- -space: S000048 -property: P000016 -value: true -refs: -- doi: 10.1007/978-1-4612-6290-9_6 - name: Counterexamples in Topology ---- - -The complement of any open set in $X$ is finite. - -See item #4 for space #56 in {{doi:10.1007/978-1-4612-6290-9_6}}. diff --git a/spaces/S000048/properties/P000208.md b/spaces/S000048/properties/P000208.md new file mode 100644 index 0000000000..4badcb7de8 --- /dev/null +++ b/spaces/S000048/properties/P000208.md @@ -0,0 +1,7 @@ +--- +space: S000048 +property: P000208 +value: true +--- + +The closed sets satisfy the descending chain condition, because all closed sets are finite except $X$. diff --git a/spaces/S000150/properties/P000208.md b/spaces/S000150/properties/P000208.md new file mode 100644 index 0000000000..2b8e2ea2b3 --- /dev/null +++ b/spaces/S000150/properties/P000208.md @@ -0,0 +1,7 @@ +--- +space: S000150 +property: P000208 +value: false +--- + +$\left[ \frac 1n, \to \right)$ is a strictly increasing sequence of open sets. diff --git a/spaces/S000151/properties/P000208.md b/spaces/S000151/properties/P000208.md new file mode 100644 index 0000000000..3cf0100e34 --- /dev/null +++ b/spaces/S000151/properties/P000208.md @@ -0,0 +1,7 @@ +--- +space: S000151 +property: P000208 +value: false +--- + +$\left( \frac 1n, \to \right)$ is a strictly increasing sequence of open sets. diff --git a/spaces/S000200/properties/P000016.md b/spaces/S000200/properties/P000016.md deleted file mode 100644 index fc6e955724..0000000000 --- a/spaces/S000200/properties/P000016.md +++ /dev/null @@ -1,7 +0,0 @@ ---- -space: S000200 -property: P000016 -value: true ---- - -Any open cover must contain $\omega$. diff --git a/spaces/S000200/properties/P000208.md b/spaces/S000200/properties/P000208.md new file mode 100644 index 0000000000..f2a1d37a88 --- /dev/null +++ b/spaces/S000200/properties/P000208.md @@ -0,0 +1,7 @@ +--- +space: S000200 +property: P000208 +value: true +--- + +All closed sets are left rays, so every closed set except for $\omega$ is finite. This implies the descending chain condition on closed sets. diff --git a/theorems/T000251.md b/theorems/T000251.md index b45c574f27..eeafd73717 100644 --- a/theorems/T000251.md +++ b/theorems/T000251.md @@ -3,7 +3,10 @@ uid: T000251 if: P000129: true then: - P000208: true + P000016: true +refs: + - mathse: 3844039 + name: What topological properties are trivially/vacuously satisfied by any indiscrete space? --- -The ascending chain condition on open sets holds since there are at most two open sets. +All open covers are finite to begin with. diff --git a/theorems/T000651.md b/theorems/T000651.md new file mode 100644 index 0000000000..5e9d86b62e --- /dev/null +++ b/theorems/T000651.md @@ -0,0 +1,9 @@ +--- +uid: T000651 +if: + P000208: true +then: + P000041: true +--- + +See [Lemma 5.9.6](https://stacks.math.columbia.edu/tag/04MF) from the Stacks project, which makes a stronger claim. diff --git a/theorems/T000658.md b/theorems/T000658.md new file mode 100644 index 0000000000..d601a4c6fc --- /dev/null +++ b/theorems/T000658.md @@ -0,0 +1,11 @@ +--- +uid: T000658 +if: + and: + - P000016: true + - P000185: true +then: + P000208: true +--- + +The ascending chain condition on open sets holds since there are only finitely many open sets. diff --git a/theorems/T000659.md b/theorems/T000659.md new file mode 100644 index 0000000000..540b66301d --- /dev/null +++ b/theorems/T000659.md @@ -0,0 +1,13 @@ +--- +uid: T000659 +if: + and: + - P000208: true + - P000134: true +then: + P000185: true +--- + +First observe that a {P208} {P3} space is {P52}, since every subset is compact, hence closed. + +Now, if $X$ is {P208} and {P134}, its Kolmogorov quotient is {P208} and {P3}, hence {P52}. This is equivalent to $X$ being {P185}. diff --git a/theorems/T000660.md b/theorems/T000660.md new file mode 100644 index 0000000000..827c507964 --- /dev/null +++ b/theorems/T000660.md @@ -0,0 +1,13 @@ +--- +uid: T000660 +if: + and: + - P000203: true + - P000208: true +then: + P000078: true +--- + +Let $p \in X$ be the only non-isolated point. The subspace $X \setminus \{p\}$ is {P52} and {P208}, +hence {P78} [(Explore)](https://topology.pi-base.org/spaces?q=Discrete+%2B+Noetherian+%2B+%7EFinite). +Therefore $X$ is also {P78}.