An Introduction to Topology by T B Singh
Overview
Structures and Spaces in Topology
An Introduction of Topology by T B Singh
Structures and Spaces
§1 Digression on Sets
We begin with a digression, which we would like to consider unnec-
essary. Its subject is the first basic notions of the naive set theory. This
is a part of the common mathematical language, too, but even more
profound than general topology. We would not be able to say anything
about topology without this part (look through the next section to see
that this is not an exaggeration). Naturally, it may be expected that the
naive set theory becomes familiar to a student when she or he studies
Calculus or Algebra, two subjects usually preceding topology. If this is
what really happened to you, then, please, glance through this section
and move to the next one.
§1◦1 Sets and Elements
In any intellectual activity, one of the most profound actions is gath-
ering objects into groups. The gathering is performed in mind and is not
accompanied with any action in the physical world. As soon as the group
has been created and assigned a name, it can be a subject of thoughts
and arguments and, in particular, can be included into other groups.
Mathematics has an elaborated system of notions, which organizes and
regulates creating those groups and manipulating them. This system is
the naive set theory , which is a slightly misleading name because this is
rather a language than a theory.
The first words in this language are set and element. By a set we
understand an arbitrary collection of various objects. An object included
into the collection is an element of the set. A set consists of its elements.
It is also formed by them. To diversify wording, the word set is replaced
by the word collection. Sometimes other words, such as class, family , and
group, are used in the same sense, but this is not quite safe because each
of these words is associated in modern mathematics with a more special
meaning, and hence should be used instead of the word set with caution.
If x is an element of a set A, then we write x ∈ A and say that x
belongs to A and A contains x. The sign ∈ is a variant of the Greek letter
epsilon, which is the first letter of the Latin word element .