Definition:Initial Topology
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 $I$.
Definition 1
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 $\family {f_i}_{i \mathop \in I}$.
Definition 2
Let $\tau$ be the coarsest topology on $X$ such that each $f_i: X \to Y_i$ is $\tuple {\tau, \tau_i}$-continuous.
Then $\tau$ is known as the initial topology on $X$ with respect to $\family {f_i}_{i \mathop \in I}$.
Also known as
The initial topology is also known as:
- the projective topology
- the weak topology on $X$ with respect to $\family {f_i}_{i \mathop \in I}$
If only a single topological space $\struct {Y, \tau_Y}$ and a single mapping $f: X \to Y$ are under consideration, the initial topology on $X$ with respect to $f$ is additionally known as:
- the pullback topology on $X$ under $f$
- the topology on $X$ induced by $f$
- the inverse image of $\tau_Y$ under $f$
and is often denoted by $\map {f^*} {\tau_Y}$ or $\map {f^{-1} } {\tau_Y}$.
Also see
- Equivalence of Definitions of Initial Topology
- Initial Topology with respect to Mapping equals Set of Preimages
- Domain Topology Contains Initial Topology iff Mappings are Continuous
- Definition:Final Topology
- Results about the initial topology can be found here.