0.1 Topological Spaces
On non-empty set \(X\) a topology \(\mathcal{T}\) is a collection of subsets of \(X\) that satisfies the following: 1. \(\emptyset, X \in \mathcal{T}\) 2. \(\{ U_{\alpha} \}_{\alpha \in A}\subset \mathcal{T}\implies \bigcup_{\alpha \in A}U_{\alpha}\in \mathcal{T}\) (closed under arbitrary unions) 3. \(\{ U_{i} \}_{i=1}^n\subset\mathcal{T}\implies \bigcap_{i=1}^n U_{i}\in\mathcal{T}\) (closed under finite intersections) The pair \((X,\mathcal{T})\) is known as a topological space.
Some Examples: - If \(X\) is a [[knowledge-mathematics-analysis-functional-analysis-metric-spaces|metric space]], the collection of all open sets with respect to the metric is a topology on \(X\). - If \((X,\mathcal{T})\) is a topological space and \(Y\subset X\), then \(\mathcal{T}_{Y}=\{ U \cap Y:U \in \mathcal{T} \}\) is a topology on \(Y\), called the relative topology induced by \(\mathcal{T}\).
0.1.0.1 Key Terminology
Open & Closed Sets:
The members of \(\mathcal{T}\) are called open sets and their complements are called closed sets. If \(Y \subset X\), the open (closed) subsets of \(Y\) in the relative topology are called relatively open (closed). By De Morgan's Laws, the family of closed sets is closed under arbitrary intersections and finite unions.
Interior & Closure: For \(A\subset X\), the union of all open sets contained in \(A\) is called the interior of \(A\), denoted \(A^0\). The intersection of all closed sets containing \(A\) is called the closure of \(A\), denoted \(\overline{A}\). The interior \(A^0\) is the largest open set contained in \(A\) and the closure \(\overline{A}\) is the smallest closed set containing \(A\). Furthermore, we have that \((A^0)^c=\overline{A^c}~~\&~~(\overline A)^c=(A^c)^0.\)
Boundary: The boundary of \(A\) is defined as the difference \(\delta A:=\overline{A}\setminus A^0=\overline{A}\cap \overline{A^c}\).
Density: If \(\overline A = X\) then \(A\) is called dense in \(X\). Alternatively, if \((\overline{A})^o=\emptyset\) then \(A\) is called nowhere dense.
Neighborhood: If \(x \in X\) (or \(E\subset X\)), a neighborhood of \(x\) or \(E\) is a set \(A\subset X\) such that \(x \in A^o\) or (\(E\subset A^o\)). This, a set \(A\) is open iff it is a neighborhood of itself.
Accumulation Point: A point \(x\) is called an accumulation point of \(A\) if \(A\cap(U \setminus \{ x \})\neq \emptyset\) for every neighborhood \(U\) of \(x\).