Topology

Topology on a set \( X \)

A topology on \( X \) is a collection \( \mathcal{T} \) of subsets of \( X \), that is: it is a subset of the \( P \left ( X \right ) \), with the following properties:

  1. \( \emptyset , X \in \mathcal{T} \)
  2. Suppose \( \left \lbrace U_{\alpha} \right \rbrace \) is a family of sets in \( \mathcal{T} \) then
    3. \( \bigcup_{\alpha} U_{\alpha} \)\( \in \mathcal{T} \)
  3. Suppose \( \left \lbrace U_{i} \right \rbrace_{i = 1}^{n} \) is a finite family of set in \( \mathcal{T} \) then
    4. \( \bigcap_{i = 1}^{n} U_{i} \) \( \in \mathcal{T} \)
The set \( X \) along with \( \mathcal{T} \) satsifying the above conditions is called a topological space and is denoted by \( \left ( X , \mathcal{T} \right ) \)
Open set
Suppose \( \left ( X , \mathcal{T} \right ) \) is a topological space, if \( U \in \mathcal{T} \) then we say that \( U \) is open with respect to \( X \).
a set filled with open sets is open
Let \( X \) be a topological space, and \( A \subseteq X \). Suppose that for each \( x \in A \) there is an open set \( U \) containing \( x \) such that \( U \subseteq A \). Show that \( A \) is open in \( X \).

Since for each \( x \in A \), there is an open set \( U_{x} \) such that \( x \in U_{x} \subseteq A \), then \( A \) is covered by subsets, therefore it is a union, and we can write \( A = \bigcup_{x \in A} U_{x} \)

Since each \( U_{x} \) was assumed to be open with respect to \( X \), then an arbitrary union of them is also open with respect to \( X \), in other words \( A \) must be open.

finer and coarser topologies

suppose that \( \mathcal{T} \) and \( \mathcal{T} ' \) are two topologies on a given set \( X \). If \( \mathcal{T} \subseteq \mathcal{T} ' \), then \( \mathcal{T} ' \) is finer than \( \mathcal{T} \). If the reverse inclusion is true, then we say that \( \mathcal{T} ' \) is coarser than \( \mathcal{T} \), there are also strict variations of these definitions for the strict inclusions.

comparable topologies

given two topologies, they are comparable if at least one is finer than the other