# Final Topology with respect to Mapping

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Although this article appears correct, it's inelegant. There has to be a better way of doing it.In particular: The presentation of the material on this page is difficult to follow. I believe there is probably a clearer way to present this.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Improve}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Theorem

Let $\struct {X, \tau_X}$ be a topological space.

Let $Y$ be a set.

Let $f: X \to Y$ be a mapping.

Let $\tau_Y$ be the final topology on $Y$ with respect to $f$.

Then:

- $\tau_Y = \set {U \subseteq Y : f^{-1} \sqbrk U \in \tau_X}$

Observe that the set on the right hand side of the equality is sometimes denoted $f\tau$.

Further, the following holds true:

- $\forall \struct {Z, \tau_Z}$ topological space $\forall g: \struct {Y, f \tau} \to \struct {Z, \tau_Z}$ mapping $g$ is continuous $\iff g \circ f$ is continuous (where $f$ is defined from $\struct {X, \tau_X}$ to $\struct {Y, f \tau}$).
- $\forall \tau_Y$ topology in $Y : \forall \struct {Z, \tau_Z}$ topological space, the following are equivalent:
- $\tau_Y = f \tau$
- $\forall g: \struct {Y, \tau_Y} \to \struct {Z, \tau_Z}: g$ is continuous $\iff g \circ f$ is continuous (where $f$ is defined from $\struct {X, \tau_X}$ to $\struct {Y, \tau_Y}$).

That is, the topology $f \tau$ is *characterised* by the fact that it is the only one which makes all mappings defined from it to any topological space continuous.

## Proof

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |