Topology

Topology on a set

X

A topology on $X$ is a collection $T$ of subsets of $X$ , that is: it is a subset of the $P (X)$ , with the following properties:

$\emptyset, X \in T$
Suppose ${U_{α}}$ is a family of sets in $T$ then

⋃_{α} U_{α}

\in T

Suppose ${U_{i}}_{i = 1}^{n}$ is a finite family of set in $T$ then

⋂_{i = 1}^{n} U_{i}

\in T

The set

X

along with

T

satsifying the above conditions is called a topological space and is denoted by

(X, T)

Open set

Suppose

(X, T)

is a topological space, if

U \in 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 $A = ⋃_{x \in A} U_{x}$

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 $T$ and $T^{'}$ are two topologies on a given set $X$ . If $T \subseteq T^{'}$ , then $T^{'}$ is finer than $T$ . If the reverse inclusion is true, then we say that $T^{'}$ is coarser than $T$ , 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