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+---
+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
+
+