The following are known as separation axioms for a topological space \(\tuple{X,\mc T}\text{.}\)
\(\mc T\) is said to be \(T_0\) if and only if for all points \(x,y\in X\) such that \(x\not=y\text{,}\) there either exists a neighborhood \(U\) of \(x\) such that \(y\not\in U\text{,}\) or there exists a neighborhood \(V\) of \(y\) such that \(x\not\in V\text{.}\)
\(\mc T\) is said to be \(T_1\) if and only if for all points \(x,y\in X\) such that \(x\not=y\text{,}\) there exists a neighborhood \(U\) of \(x\) such that \(y\not\in U\text{.}\)
\(\mc T\) is said to be \(T_2\) (also known as Hausdorff) if and only if for all points \(x,y\in X\) such that \(x\not=y\text{,}\) there exist neighborhoods \(U,V\) of \(x,y\) (respectively) such that \(U\cap V=\emptyset\text{.}\)
Proposition4.2.
\(T_2\Rightarrow T_1\Rightarrow T_0\text{.}\)
Checkpoint4.3.
Find or create an example of a topological space \(\tuple{X,\mc T}\) that is:
Not \(T_0\text{.}\)
\(T_0\) but not \(T_1\text{.}\)
\(T_1\) but not \(T_2\text{.}\)
Theorem4.4.
A topological space \(X\) is \(T_1\) if and only if every finite subset of \(X\) is closed.
Corollary4.5.
Let \(X\) be a finite topological space. Then \(X\) is \(T_1\) if and only if \(X\) has the discrete topology.
Proposition4.6.
The Euclidean real line is an example of a Hausdorff topological space that does not have the discrete topology.
Definition4.7.
The following are also known as separation axioms for a topological space \(\tuple{X,\mc T}\text{.}\)
\(\mc T\) is said to be regular if and only if for all points \(x\in X\) and closed subsets \(H\subseteq X\) such that \(x\not\in H\text{,}\) there exist open sets \(U,V\in\mc T\) such that \(x\in U,H\subseteq V,U\cap V=\emptyset\text{.}\)
\(\mc T\) is said to be \(T_3\) if and only if it is both regular and \(T_1\)
\(\mc T\) is said to be normal if and only if for all closed subsets \(H,L\subseteq X\) such that \(H\cap L=\emptyset\text{,}\) there exist open sets \(U,V\in\mc T\) such that \(H\subseteq U,L\subseteq V,U\cap V=\emptyset\text{.}\)
\(\mc T\) is said to be \(T_4\) if and only if it is both normal and \(T_1\)
Proposition4.8.
\(T_{n+1}\Rightarrow T_n\) for \(n\in\setList{0,1,2,3}\text{.}\)
Theorem4.9.
The real line \(\mb R\) equipped with the Euclidean topology is \(T_3\text{.}\)
\(T_4\text{,}\)
Checkpoint4.10.
Find or create an example of a topological space that is:
\(T_2\) but not regular.
\(T_3\) but not \(T_4\)
Regular but not \(T_3\text{.}\)
Normal but not \(T_4\text{.}\)
Regular but not normal.
Normal but not regular.
Proposition4.11.
A space is regular if and only if for every point \(x\) and neighborhood \(U\text{,}\) there exists a neighborhood \(V\) of \(x\) such that \(x\in V\subseteq\cl V\subseteq U\text{.}\)
Proposition4.12.
A space is normal if and only if for closed set \(H\) and open set \(U\supseteq H\text{,}\) there exists an open set \(V\supseteq H\) such that \(H\subseteq V\subseteq\cl V\subseteq U\text{.}\)
Theorem4.13.
A topological space is \(T_3\) if and only if it is regular and \(T_0\text{.}\)
Proposition4.14.
Let \(n\in\setList{0,1,2,3}\text{.}\) A subspace of a \(T_n\) space is also \(T_n\text{.}\)
Proposition4.15.
Let \(n\in\setList{0,1,2}\) and \(f:X\to Y\) be continuous. If \(Y\) is \(T_n\text{,}\) then \(X\) is also \(T_n\text{.}\)
Theorem4.16.
Let \(n\in\setList{3,4}\) and \(f:X\to Y\) be a continuous closed map. If \(Y\) is \(T_n\text{,}\) then \(X\) is also \(T_n\text{.}\)
Corollary4.17.
Separation properties are topological properties (that is, preserved by homeomorphism).
