Various definitions in set theory
date
Oct 2, 2024
lang
en
slug
various-definitions-in-set-theory
status
Published
type
Note
url
🦋 brain
Text
Overview
Set theory is recognized as a fundamental pillar of modern mathematics. A more recent foundational framework in modern mathematics is category theory. Within set theory, the concept of general topology is used to define and understand the notion of "nearness".
Topological space
Definition
In this case,
We call
- : open set
- : general topology
Example 1: Discrete space
Consider a set . The discrete topology on is defined by letting be the power set of , i.e., . This means that every subset of is an open set in this topology.
For example, the open sets in this topology include:
This satisfies the conditions for a topology:
- Both and are in : This is evident as both are listed.
- The intersection of any finite number of open sets is open: For instance, , which is open.
- The union of any collection of open sets is open: For example, , which is open.
Example 2: Indiscrete space
Consider the same set . The indiscrete topology (codiscrete topology) on is defined by letting . This means that only the empty set and the entire set are open in this topology.
This also satisfies the conditions for a topology:
- Both and are in : This is evident by definition.
- The intersection of any finite number of open sets is open: Since the only open sets are and , any intersection will result in one of these two sets, which are both open.
- The union of any collection of open sets is open: Since the only open sets are and , any union will result in , which is open.
Continuous function
Definition
In this case,
Compact space
Definition
In this case,