Category:Definitions/Product Topology
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This category contains definitions related to Product Topology.
Related results can be found in Category:Product Topology.
Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.
Let $\XX$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$:
- $\ds \XX := \prod_{i \mathop \in I} X_i$
For each $i \in I$, let $\pr_i: \XX \to X_i$ denote the $i$th projection on $\XX$:
- $\forall \family {x_j}_{j \mathop \in I} \in \XX: \map {\pr_i} {\family {x_j}_{j \mathop \in I} } = x_i$
The product topology on $\XX$ is defined as the initial topology $\tau$ on $\XX$ with respect to $\family {\pr_i}_{i \mathop \in I}$.
That is, $\tau$ is the topology generated by:
- $\SS = \set {\pr_i^{-1} \sqbrk U: i \in I, U \in \tau_i}$
where $\pr_i^{-1} \sqbrk U$ denotes the preimage of $U$ under $\pr_i$.
Pages in category "Definitions/Product Topology"
The following 17 pages are in this category, out of 17 total.
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- Definition:Product Space (Topology)
- Definition:Product Space (Topology) of Two Factor Spaces
- Definition:Product Space (Topology)/Two Factor Spaces
- Definition:Product Space of Topological Spaces
- Definition:Product Topology
- Definition:Product Topology on Finite Product
- Definition:Product Topology on Two Factor Spaces
- Definition:Product Topology/Also known as
- Definition:Product Topology/Factor Space
- Definition:Product Topology/Finite Product
- Definition:Product Topology/Natural Basis
- Definition:Product Topology/Natural Sub-Basis
- Definition:Product Topology/Two Factor Spaces