3.1 Definitions

Definition 3.1.1 (Topological Space).label Let $X$ be a set, possibly empty. A topology on $X$ is a structure given by a collection $\tau\suf\mathcal{P}(X)$ satisfying:

1. (1)

   $\emp,X\in \tau$

2. (2)

   $\fall \curl{U_i}_{i\in I}\suf \tau,\cups{}{i\in I}U_{i}\in \tau$

3. (3)

   $\fall U,V\in \tau, U\cap V\in \tau$

The ordered pair $(X,\tau)$ (or just $X$ if $\tau$ is understood) is a topological space and elements of $\tau$ are open sets.

The elements of a topological space are points and $X$ is the underlying set of the topological space $(X,\tau)$. A subset $C\suf X$ of a topological space is closed if $A^{c}$ is open.

Definition 3.1.2 (Closed Set).label Let $(X,\tau)$ be a topological space, then $A\suf X$ is closed if $A^{c}\in\tau$.

Definition 3.1.3 (Seperation Axioms).label Let $(X,\tau)$ be a topological space, then we have the following separation axioms which $X$ might satisfy

1. (T0)

   For any $x\neq y\in X$ there exists $U\in \tau$ such that either $y\nin U\ni x$ or $x\nin U\ni y$

2. (T1)

   For any $x\neq y\in X$ there exists $U\in\tau$ such that $y\nin U\ni x$

3. (T2)

Definition 3.1.4 (Covering).label Let $A\suf X$ be subset of a topological space. A collection $(U_{i})_{i\in I}$ is called a covering of $A$ if $A\suf\cups{}{i\in I}U_{i}$ and $U_{i}\suf X$ for each $i\in I$. The covering is said to be open if all $U_{i}$ are open in $X$.