Topologies and Topological Spaces

Topologies and Topological Spaces

Definition: Let $E$ be a set. A collection $\tau$, of subsets of $E$ is a Topology on $E$ if it satisfies the following three conditions: (1) $E \in \tau$ and $\emptyset \in \tau$. (2) Every union of an arbitrary collection of sets in $\tau$ is also contained in $\tau$. (3) Every intersection of a finite collection of sets in $\tau$ is also contained in $\tau$. The pair $(E, \tau)$ is called a Topological Space.

Open Sets and Closed Sets

Definition: Let $(E, \tau)$ be a topological space. A subset $A \subseteq E$ is said to be an Open Set if $E \in \tau$, and a subset $B \subseteq E$ is said to be a Closed Set if its complement, $E^c = E \setminus B$ is an open set.

Note that there exists sets that are both open and closed (for example, the whole space $E$, and $\emptyset$). Also note that there exists sets that are neither open or closed.

An infinite intersection of open sets need not be open. For example, $\left (0, 1 + \frac{1}{n} \right )$ is an open set for each $n \in \mathbb{N}$ but:

(1)

\begin{align} \bigcap_{n=1}^{\infty} \left ( 0, 1 + \frac{1}{n} \right ) = (0, 1] \end{align}

is not an open set.

Proposition 1: Let $(E, \tau)$ be a topological space. Then: (1) $E$ and $\emptyset$ are closed sets. (2) Every union of a finite collection of closed sets is a closed set. (3) Every intersection of an arbitrary collection of closed sets is a closed set.

An infinite union of closed sets need not be closed. For example, $\left [ 0, 1 - \frac{1}{n} \right ]$ is a closed set for each $n \in \mathbb{N}$ but:

(2)

\begin{align} \bigcup_{n=1}^{\infty} \left [ 0, 1 - \frac{1}{n} \right ] = [0, 1) \end{align}

is not a closed set.

Neighbourhoods of Points

Definition: Let $(E, \tau)$ be a topological space. A subset $U \subseteq E$ is a Neighbourhood of a point $x \in E$ if there exists an open set $V$ with $x \in V \subseteq U$.

Proposition 2: Let $(E, \tau)$ be a topological space and let $x \in E$. Then: (1) $x \in U$ for all neighbourhoods $U$ of $x$. (2) If $U$ and $V$ are neighbourhoods of $x$ then $U \cap V$ is a neighbourhood of $x$. (3) If $U$ is a neighbourhood of $x$ and $U \subseteq V$ then $V$ is a neighbourhood of $x$. (4) If $U$ is a neighbourhood of $x$ then there exists a neighbourhood of $x$, call it $V$, such that $U$ is a neighbourhood of $y$ for all $y \in V$. Conversely, if $E$ is a set such that for each $x \in E$ there exists a collection $\mathcal U_x$ of subsets of $E$ with the properties that: (1*) $x \in U$ for each $U \in \mathcal U_x$. (2*) If $U, V \in \mathcal U_x$ then $U \cap V \in \mathcal U_x$. (3*) If $U \in \mathcal U_x$ and $U \subseteq V$ then $V \in \mathcal U_x$. (4*) For all $U \in \mathcal U_x$ there exists a $V \in \mathcal U_x$ such that $U \in \mathcal U_y$ for each $y \in V$. Then there exists a unique topology on $E$ such that for each $x \in E$, $\mathcal U_x$ is a set of neighbourhoods of $x$.

Hausdorff Topological Spaces

Definition: A topological space $(E, \tau)$ is a Hausdorff Topological Space if for every pair of distinct points $x, y \in E$, there exists neighbourhoods $U$ or $x$ and $V$ of $y$ such that $U \cap V = \emptyset$.

Comparable Topologies

Definition: Let $(E, \tau_1)$ and $(E, \tau_2)$ be topological spaces on the same set $E$. Then $\tau_1$ is Finer than $\tau_2$ if $\tau_2 \subseteq \tau_1$, or in other words, every set that is open in $(E, \tau_2)$ is also open in $(E, \tau_1)$. If either $\tau_1$ is finer than $\tau_2$ or $\tau_2$ is finer than $\tau_1$ then the topologies $\tau_1$ and $\tau_2$ are said to be Comparable Topologies.