Category:Topological Bases
This category contains results about Topological Bases in the context of Topology.
Definitions specific to this category can be found in Definitions/Topological Bases.
Analytic Basis
Let $\left({S, \tau}\right)$ be a topological space.
An analytic basis for $\tau$ is a subset $\mathcal B \subseteq \tau$ such that:
- $\displaystyle \forall U \in \tau: \exists \mathcal A \subseteq \mathcal B: U = \bigcup \mathcal A$
That is, such that for all $U \in \tau$, $U$ is a union of sets from $\mathcal B$.
Synthetic Basis
Let $S$ be a set.
Definition 1
A synthetic basis on $S$ is a subset $\mathcal B \subseteq \mathcal P \left({S}\right)$ of the power set of $S$ such that:
| \((B1)\) | $:$ | $\mathcal B$ is a cover for $S$ | ||||||
| \((B2)\) | $:$ | \(\displaystyle \forall U, V \in \mathcal B:\) | $\exists \mathcal A \subseteq \mathcal B: U \cap V = \bigcup \mathcal A$ |
That is, the intersection of any pair of elements of $\mathcal B$ is a union of sets of $\mathcal B$.
Definition 2
A synthetic basis on $S$ is a subset $\mathcal B \subseteq \mathcal P \left({S}\right)$ of the power set of $S$ such that:
- $\mathcal B$ is a cover for $S$
- $\forall U, V \in \mathcal B: \forall x \in U \cap V: \exists W \in \mathcal B: x \in W \subseteq U \cap V$
Subcategories
This category has the following 2 subcategories, out of 2 total.
Pages in category "Topological Bases"
The following 30 pages are in this category, out of 30 total.
B
E
- Equivalence of Definitions of Analytic Basis
- Equivalence of Definitions of Synthetic Basis
- Equivalence of Definitions of Topology Generated by Synthetic Basis
- Equivalence of Definitions of Topology Generated by Synthetic Basis/Definition 1 iff Definition 2
- Equivalence of Definitions of Topology Generated by Synthetic Basis/Definition 1 iff Definition 3
- Equivalent Definitions of Synthetic Basis
