From 28407333ffceca9b99fae721c30e8ae146a863da Mon Sep 17 00:00:00 2001 From: Justin Gassner Date: Wed, 14 Feb 2024 07:24:38 +0100 Subject: Update --- pages/general-topology/topological-spaces.md | 146 +++++++++++++++++++++++++++ 1 file changed, 146 insertions(+) create mode 100644 pages/general-topology/topological-spaces.md (limited to 'pages/general-topology/topological-spaces.md') diff --git a/pages/general-topology/topological-spaces.md b/pages/general-topology/topological-spaces.md new file mode 100644 index 0000000..b0b1834 --- /dev/null +++ b/pages/general-topology/topological-spaces.md @@ -0,0 +1,146 @@ +--- +title: Topological Spaces +parent: General Topology +nav_order: 1 +--- + +# {{ page.title }} + +## Elementary Concepts + +{% definition Topology, Topological Space %} +A *topology* on a set $X$ is a collection $\mathcal{T}$ of subsets of $X$ such that \ +**(T1)** $\varnothing$ and $X$ belong to $\mathcal{T}$, \ +**(T2)** the union of any subcollection of $\mathcal{T}$ belongs to $\mathcal{T}$, \ +**(T3)** the intersection of any finite subcollection $\mathcal{T}$ belongs to $\mathcal{T}$. \ +A *topological space* is a pair $(X,\mathcal{T})$ consisting of +a set $X$ and a topology $\mathcal{T}$ on $X$. +{% enddefinition %} + +If one follows the convention that +the union of the empty collection of subsets of $X$ is the empty subset of $X$, +and its intersection is all of $X$, +then **(T1)** is a consequence of **(T2)**, **(T3)** +and can be omitted. + +If $(X,\mathcal{T})$ is a topological space, +the elements of $X$ are called *points* +and the elements of $\mathcal{T}$ are called the *open sets*. + +{% example %} +On every set $X$ we have +the *trivial* (or *indiscrete*) *topology* $\braces{\varnothing,X}$ and +the *discrete topology* $\mathcal{P}(X)$. +These collections are in fact topologies on $X$. +{% endexample %} + +{% example %} +If $X$ is any set, +then the collection of all subsets of $X$ +whose complement is either finite or all of $X$ +is a topology on $X$; +it is called the *finite complement topology*. +The *countable complement topology* is defined analogously. +{% endexample %} + +{% definition Comparison of Topologies %} +Suppose $\mathcal{T}$ and $\mathcal{T}'$ are topologies on a set $X$. +When $\mathcal{T} \subset \mathcal{T}'$, +we say that $\mathcal{T}$ is *coarser* or *smaller* or *weaker* than $\mathcal{T}'$, +and that $\mathcal{T}'$ is *finer* or *larger* or *stronger* than $\mathcal{T}$. +If the inclusion is proper, then we say *strictly coarser* and so on. +If either $\mathcal{T} \subset \mathcal{T}'$ or $\mathcal{T} \supset \mathcal{T}'$ holds, +then the topologies are said to be *comparable*. +{% enddefinition %} + +{% proposition Intersection of Topologies %} +If $\braces{\mathcal{T}_{\alpha}}$ is a family of topologies on a set $X$, +then $\bigcap_{\alpha} \mathcal{T}_{\alpha}$ is a topology on $X$. +{% endproposition %} + +{% definition Generated Topology %} +Suppose $\mathcal{A}$ is a collection of subsets of a set $X$. +The *topology generated by $\mathcal{A}$* is +the intersection of all topologies on $X$ containing $\mathcal{A}$. +{% enddefinition %} + +By the previous proposition, the generated topology is indeed a topology. + +{% proposition %} +The topology generated by a collection $\mathcal{A}$ of subsets of a set $X$ +is the smallest topology on $X$ containing $\mathcal{A}$. +{% endproposition %} + +## Bases and Subbases + +{% definition Basis for a Topology %} +A *basis for a topology* on a set $X$ is a collection $\mathcal{B}$ of subsets of $X$ +such that for every point $x \in X$ +- there exists $B \in \mathcal{B}$ such that $x \in B$, +- if $x \in B_1 \cap B_2$ for $B_1, B_2 \in \mathcal{B}$, + then there exists a $B_3 \in \mathcal{B}$ + such that $x \in B_3 \subset B_1 \cap B_2$. +{% enddefinition %} + +{% theorem Topology Generated by a Basis %} +If $X$ is set and $\mathcal{B}$ is a basis for a topology on $X$, +then the topology generated by $\mathcal{B}$ equals +- the collection of all subsets $S \subset X$ with the property + that for every $x \in S$ there exists a basis element $B \in \mathcal{B}$ + such that $x \in B$ and $B \subset S$; +- the collection of all arbitrary unions of elements of $\mathcal{B}$. +{% endtheorem %} + +Let $\mathcal{T}$ be a topology on a set $X$. +As one might expect, +a collection $\mathcal{B}$ of subsets of $X$ +is said to be a *basis for the topology $\mathcal{T}$*, +if $\mathcal{B}$ is basis for a topology on $X$ and +the topology generated by $\mathcal{B}$ equals $\mathcal{T}$. + +{% example %} +If $X$ is a set, then the collection of singletons $\braces{x}$, $x \in X$, +is a basis for the discrete topology on $X$. +{% endexample %} + +{% example %} +If $(X,d)$ is a metric space, +then the collection of open balls is a basis for the metric topology on $X$. +{% endexample %} + +{% definition Subbasis for a Topology %} +A *subbasis for a topology* on a set $X$ is a collection $\mathcal{S}$ of subsets of $X$ +such that for every point $x \in X$ there exists a $S \in \mathcal{S}$ such that $x \in S$. +{% enddefinition %} + +{% theorem Topology Generated by a Subbasis %} +If $X$ is set and $\mathcal{S}$ is a subbasis for a topology on $X$, +then the topology generated by $\mathcal{S}$ equals +- the collection of all arbitrary unions of finite intersections of elements of $\mathcal{S}$. +{% endtheorem %} + +## Open and Closed Sets + +{% definition Open Set, Closed Set %} +Suppose $(X,\mathcal{T})$ is a topological space. +A subset $S$ of $X$ +is called *open* with respect to $\mathcal{T}$ +when it belongs to $\mathcal{T}$ +and it is called *closed* with respect to $\mathcal{T}$ +when its complement $X \setminus S$ belongs to $\mathcal{T}$. +{% enddefinition %} + +A subset of a topological space is open +if and only if its complement is closed. + +{% proposition %} +Let $\mathcal{C}$ be the collection of closed subsets of a topological space. Then +{: .mb-0 } +- $X$ and $\varnothing$ belong to $\mathcal{C}$, +- the intersection of any subcollection of $\mathcal{C}$ belongs to $\mathcal{C}$, +- the union of any finite subcollection $\mathcal{C}$ belongs to $\mathcal{C}$. +{% endproposition %} + +## The Subspace Topology + + -- cgit v1.2.3-54-g00ecf