Sub-Basis for Initial Topology in terms of Sub-Bases of Target Spaces

Theorem

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 $I$.

For each $i \in I$, let $S_i$ be a synthetic basis for $\struct {Y_i, \tau_i}$.

Let $\family {f_i: X \to Y_i}_{i \mathop \in I}$ be an indexed family of mappings indexed by $I$.

Let $\tau$ be the initial topology on $X$ with respect to $\family {f_i}_{i \mathop \in I}$.

Then:

$\SS = \set {f_i^{-1} \sqbrk U : i \in I, \, U \in S_i}$

is a synthetic sub-basis for $\struct {X, \tau}$.

Proof

Note that by the definition of the initial topology, $\tau$ is generated by the synthetic sub-basis:

$\SS' = \set {f_i^{-1} \sqbrk U : i \in I, \, U \in \tau_i}$

Since $S_i \subseteq \tau_i$ for each $i \in I$, we have:

$\SS \subseteq \SS'$

and hence:

$\map \tau \SS \subseteq \map \tau {\SS'} = \tau$

where $\tau$ denotes the generated topology.

Now fix $i \in I$ and $U \in \tau_i$.

Since $S_i$ is a synthetic sub-basis for $\tau_i$, there exists an indexing set $A$ such that:

for each $\alpha \in A$ there exists $n_\alpha \in \N$ and $U_{\alpha, 1}, U_{\alpha, 2}, \ldots, U_{\alpha, n_\alpha}$ such that together:

$\ds U = \bigcup_{\alpha \mathop \in A} \, \bigcap_{i \mathop = 1}^{n_\alpha} U_{\alpha, n_\alpha}$

Then we have:

\(\ds f_i^{-1} \sqbrk U\)

\(=\)

\(\ds f_i^{-1} \sqbrk {\bigcup_{\alpha \mathop \in A} \, \bigcap_{i \mathop = 1}^{n_\alpha} U_{\alpha, n_\alpha} }\)

\(\ds \)

\(=\)

\(\ds \bigcup_{\alpha \mathop \in A} f_i^{-1} \sqbrk {\bigcap_{i \mathop = 1}^{n_\alpha} U_{\alpha, n_\alpha} }\)

Preimage of Union under Mapping

\(\ds \)

\(=\)

\(\ds \bigcup_{\alpha \mathop \in A} \, \bigcap_{i \mathop = 1}^{n_\alpha} f_i^{-1} \sqbrk {U_{\alpha, n_\alpha} }\)

Preimage of Intersection under Mapping

where $f_i^{-1} \sqbrk {U_{\alpha, n_\alpha} } \in \SS$ for each $\alpha \in A$.

So $f_i^{-1} \sqbrk U$ can be expressed as the set union of finite intersections of elements of $\map \tau \SS$.

Since topologies are closed under unions and finite intersections, for each $i \in I$ and $U \in \tau_i$ we have $f_i^{-1} \sqbrk U \in \map \tau \SS$.

So we have shown that:

$\SS' \subseteq \map \tau \SS$

so that:

$\tau = \map \tau {\SS'} \subseteq \map \tau {\SS'}$

giving:

$\map \tau \SS = \map \tau {\SS'} = \tau$

So $\SS$ is a synthetic sub-basis for $\struct {X, \tau}$.

$\blacksquare$