English literature - basic notions

[Martin Sleziak]

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

• 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.

Base for a topology

• 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]

Some topological spaces

• Sorgenfrey line (Lower limit topology): [Engelking, Example 1.2.2], [Willard, Problem 4A]
• Moore plane: [Engelking, Example 1.2.4], [Willard, Problem 4B]

Subbase for a 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

• Definition of a neighborhood: [Engelking, p.12], [Willard, Definition 4.1]
• Definition of a neighborhood base: [Engelking, p.12], [Willard, Definition 4.3]

Our notation: $\mathcal N_x$ is the set of all neighborhoods of $x$ and $\mathcal O_x$ is the set of all neighborhoods of $x$

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]

Dense set

• 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]