Definition:Initial Topology/Definition 1
Jump to navigation
Jump to search
Definition
Let $X$ be a set.
Let $I$ be an indexing set.
Let $\family {\struct {Y_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces indexed by $I$.
Let $\family {f_i: X \to Y_i}_{i \mathop \in I}$ be an indexed family of mappings indexed by $I$.
Let:
- $\SS = \set {f_i^{-1} \sqbrk U: i \in I, U \in \tau_i}$
where $f_i^{-1} \sqbrk U$ denotes the preimage of $U$ under $f_i$.
The topology $\tau$ on $X$ generated by $\SS$ is called the initial topology on $X$ with respect to $\family {f_i}_{i \mathop \in I}$.
Also see
- Results about the initial topology can be found here.
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.3$: Sub-bases and weak topologies