Some cardinal estimations via the inclusion-exclusion principle in finite \(T_0\) topological spaces. (English) Zbl 07746967
Acharjee, Santanu (ed.), Advances in topology and their interdisciplinary applications. Singapore: Springer. Ind. Appl. Math., 103-108 (2023).
After introductory discussions of interval arithmetic and point-set topology, ten results are presented. Each paraphrasing below assumes an arbitrary finite topological space \(X\).
However, the reviewer would like to note:
For the entire collection see [Zbl 1515.54002].
- (1)
- If \(|\overline{\{a\}}|\leq j\) and \(|\overline{\{b\}}|\leq k\) then \(|\overline{\{a,b\}}|\leq j+k-|\overline{\{a\}}\cap\overline{\{b\}}|\).
- (2)
- If also \(\overline{\{a\}}\cap\overline{\{b\}}=\emptyset\) in \((1)\) then \(2\leq|\overline{\{a,b\}}|\leq j+k\).
- (3)
- If \(|\overline{\{a\}}|\leq j\) and \(\overline{\{b\}}\subseteq\overline{\{a\}}\) then \(|\overline{\{a,b\}}|\leq j\).
- (4)
- If \(\overline{\{b\}}\subseteq\overline{\{a\}}\) then \(|\overline{\{a,b\}}|=|\overline{\{a\}}|\).
- (5)
- If \(X=A\cup B\) and \(A\cap B=\emptyset\) then \(|X|=|A|+|B|\).
- (6)
- If \(|\{a,b\}^\circ|\leq1\) then \(\{a\}^\circ=\{b\}^\circ=\emptyset\).
- (7)
- If \(|\{a,b\}^\circ|=2\) then \(|\{a\}^\circ|\leq1\) and \(|\{b\}^\circ|\leq1\).
- (8)
- If \(\overline{\{a\}}\cap\overline{\{b\}}=\emptyset\) and \(2\leq\overline{\{a\}}\) for all \(a\neq b\) in \(A\) then \(|\overline{A}|\geq2|A|\).
- (9)
- If also \(|\overline{\{a_j\}}|\leq k_j\) for all \(a_j\in A\) in \((8)\) then \(|X|-\sum_{j=1}^{|A|}k_j\leq|(X\setminus A)^\circ|\leq|X|-2|A|\).
- (10)
- If \(\emptyset\subsetneq B\subsetneq A\) then for some \(k\geq1\) we have \(k+1\leq|A|\leq2k-1\).
However, the reviewer would like to note:
- ●
- Some theorems impose conditions not needed in the proof (\(T_0\) space (1–4, 8–10), open subset (5, 10), \(|\{a,b\}^\circ|=2\) (7)).
- ●
- Some theorems are false ((6) is false for \(|\{a,b\}^\circ|=1\) and (10) is false for \(|A|=2\)).
- ●
- Most of the 17 listed references have little or nothing to do with the paper.
For the entire collection see [Zbl 1515.54002].
Reviewer: Mark Bowron (Las Vegas)
MSC:
| 54A05 | Topological spaces and generalizations (closure spaces, etc.) |
| 05A99 | Enumerative combinatorics |
References:
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