Topological space

A topological space is a fundamental mathematical structure that generalizes the concept of geometrical spaces and continuity. A topological space is equipped with a collection of open sets, capturing the intuitive idea of "nearness" without necessarily defining a metric. Topological spaces are the objects of study in general topology.

Definition

An ordered pair $(X,\tau )$ is a topological space on set $X$, if $\tau \subseteq 𝒫(X)$ satisfies the following properties:

• $X,\varnothing \in \tau$,
• if $𝒰\subseteq \tau$, then $\bigcup _{U\in 𝒰}U\in \tau$,
• if $U_1,U_2,\cdots ,U_n\in \tau$, then $\bigcap _{i=1}^n{U}_i\in \tau$.

Elements of $\tau$ are called open sets.

Examples

Standard topology

The real line $ℝ$ equipped with the standard topology ${\tau }_{ℝ}$ is a topological space.

The standard topology on $ℝ$ is defined by taking all open intervals as a basis. A set $U\subseteq ℝ$ is open, if for all point $x\in U$, there exists an open interval $(a,b)$ such that $x\in (a,b)\subseteq U$.

Proof

• $\varnothing \in {\tau }_{ℝ}$ vacuously; every point $x\in ℝ$ belongs to some open interval, like $(x-1,x+1)$, which is open in $ℝ$. Therefore by the definition, $ℝ\in {\tau }_{ℝ}$.
• Let $\{U_i{\}}_{i\in I}$ be open sets and $U=\bigcup _{i\in I}{U}_i$. Take $x\in U$, then $x\in U_i$ for some $i\in I$. Because $U_i$ is open, there exists $(a,b)$ such that $x\in (a,b)\subset U_i\subset U$. Therefore $U$ is open.
• The finite intersection property can be proved by induction. Let $U,V$ be open and $x\in U\cap V$. By definition of openness, there exists $(a_1,b_1)\subset U$ and $(a_2,b_2)\subset V$ such that $(a_1,b_1)\ni x\in (a_2,b_2)$. Set $a=\max \{a_1,a_2\}$ and $b=\min \{b_1,b_2\}$. Then $x\in (a,b)\subset (a_1,b_1)\cap (a_2,b_2)\subset U\cap V$. Thus $U\cap V$ is open. By induction, ${\displaystyle }$

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Basis

• for every $x\in X$, there exists $B\in ℬ$ with $x\in B$,
• if $x\in B_1\cap B_2$ with $B_1,B_2\in ℬ$, then there exists $B_3\in ℬ$ such that $x\in B_3\subseteq B_1\cap B_2$.

The topology generated by $ℬ$ consists of all unions of elements of $ℬ$.

Topological properties

Some key properties of topological spaces include:

• Compactness
• Connectedness
• Hausdorff property
• Second countability