Royal Road to Topology Chapter II
Chapter 2: From Convergence of Sequences to the Concept of Filter
Section titled “Chapter 2: From Convergence of Sequences to the Concept of Filter”This chapter traces the transition from the classical convergence of sequences to the more general and powerful concept of filters.
II.1. Convergence of Sequences
Section titled “II.1. Convergence of Sequences”- Sequence (点列・数列)
- Cofinite Subset (余有限集合)
- Neighborhood of $x$ (近傍) (in $\mathbb{R}$)
- Neighborhood Filter Properties
- Filter of Cofinite Preimages ($\Gamma_\varphi$)
- Subsequence (部分列)
- Correlation (相関)
II.2. The Concept of Filter
Section titled “II.2. The Concept of Filter”II.3. Order on Filters
Section titled “II.3. Order on Filters”- Order (包含関係による順序)
- Fréchet Filter (フレシェ・フィルター) (Definition II.3.8)
- Restriction of a Filter (フィルターの制限)
- Filter Complementary Set Extension ($\mathcal{F} \setminus A$)
- Ultrafilter (極大フィルター / ウルトラフィルター) (Definition II.3.9)
II.4. Decomposition into Free and Principal Filters
Section titled “II.4. Decomposition into Free and Principal Filters”- Kernel (核)
- Free Part (自由部分)
- Filter Decomposition Theorem (Theorem II.4.1)
- Cofinite Filter of $B$ centered at $A$ (Definition II.4.3)