Topologies and Topological Spaces

# 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. |