English literature - basic notions
Posted: Wed Oct 02, 2024 11:52 am
The text on the course website is in Slovak - this is mainly for the students who study in English.
I will list some places where some of the topics discussed in the lecture can be found.
I will mostly use Willard's General Topology and Engelking's General Topology (1989 edition). Naturally, the same definitions and results can be found in many other sources.
Closed sets
Topology determined by closed sets: [Engelking, Proposition 1.2.5], [Willard, Theorem 3.4]
Cofinite topology and cocountable topology
Generating a topology from a subbase: [Engelking, Exercise 1.2.A], [Willard, Theorem 5.6]
Notice that the convention in Willard is that the intersection of the empty system is $X$. We use different convention - which is why we require $\bigcup\mathcal S=X$ while [Willard, Theorem 5.6] says that any system can be a subbase.
Neighborhood, neighborhood basis
Interior and closure
I will list some places where some of the topics discussed in the lecture can be found.
I will mostly use Willard's General Topology and Engelking's General Topology (1989 edition). Naturally, the same definitions and results can be found in many other sources.
Closed sets
Topology determined by closed sets: [Engelking, Proposition 1.2.5], [Willard, Theorem 3.4]
Cofinite topology and cocountable topology
- Cofinite topology: [Engelking, Example 1.2.6], [Willard, Example 3.8a]
- Cocountable topology is very similar - we simply use countable subsets instead of finite subsets.
- Definition of a base for a topological space: [Engelking, p.12], [Willard, Definition 5.1]
- Generating topology from a base: [Engelking, Proposition 1.2.1], [Willard, Theorem 5.3]
- Sorgenfrey line (Lower limit topology): [Engelking, Example 1.2.2], [Willard, Problem 4A]
- Moore plane: [Engelking, Example 1.2.4], [Willard, Problem 4B]
Generating a topology from a subbase: [Engelking, Exercise 1.2.A], [Willard, Theorem 5.6]
Notice that the convention in Willard is that the intersection of the empty system is $X$. We use different convention - which is why we require $\bigcup\mathcal S=X$ while [Willard, Theorem 5.6] says that any system can be a subbase.
Neighborhood, neighborhood basis
- Definition of a neighborhood: [Engelking, p.12], [Willard, Definition 4.1]
- Definition of a neighborhood base: [Engelking, p.12], [Willard, Definition 4.3]
Interior and closure
- Definition of closure: [Willard, Definition 3.5]
- Basic properties of closure: [Engelking, Theorem 1.1.3], [Willard, Lemma 3.6 and Theorem 3.7]
- Equivalent characterization of the closure: [Engelking, Proposition 1.1.1]
- Locally finite systems and closure: [Engelking, Theorem 1.1.11 and Corollary 1.1.12]
- Definition of interior: [Willard, Definition 3.9] (some basic properties are stated immediately after the definition)
- Relationship between interior and closure: [Engelking, Theorem 1.1.5]
- Equivalent characterization of a dense set: [Engelking, Proposition 1.3.5]
- For $U$ open and $D$ dense we have $\overline{U\cap D}=\overline U$: [Engelking, Proposition 1.3.6]