# Category:Initial Topology

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This category contains results about **the initial topology**.

Definitions specific to this category can be found in Definitions/Initial Topology.

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 $\family {f_i}_{i \mathop \in I}$**.

## Subcategories

This category has the following 2 subcategories, out of 2 total.

### W

## Pages in category "Initial Topology"

The following 14 pages are in this category, out of 14 total.

### C

### E

### I

### S

- Sub-Basis for Initial Topology in terms of Sub-Bases of Target Spaces
- Subspace of Product Space has Initial Topology with respect to Restricted Projections
- Subspace Topology is Initial Topology with respect to Inclusion Mapping
- Subspace Topology on Initial Topology is Initial Topology on Restrictions