Talk:Final topology

Coarsest topology

I'm removing the following from the article:

• Given a family of topologies {τ_i} on a fixed set X the final topology on X with respect to the functions id_X : (X, τ_i) → X is the infimum (or meet) of the topologies {τ_i} in the lattice of topologies on X. That is, the final topology τ is the intersection of the topologies {τ_i}.

I beleive this one is supposed to describe the initial topology not the final topology: The intersection of two topologies will be coarser or weaker than any of the individual τ_i. Elsewhere, the article describes the final topology as being the "finest" not the "coarsest".

I'll be removing a similar paragraph from the initial topology article. linas 20:14, 20 November 2005 (UTC)

Be careful Linas; the example is correct. You are confusing greatest element with greatest lower bound. The topology on X must be coarser than each given topology to ensure continuity (i.e. a lower bound), but it is the finest topology which is coarser than each given one (i.e. a greatest lower bound). -- Fropuff 16:15, 21 November 2005 (UTC)

Right. Thanks. Sorry. linas 20:27, 25 November 2006 (UTC)

Examples

The article needs an example of a topology that satisfies conditions form being an inital toplogy, but is not a final topology. linas 20:14, 20 November 2005 (UTC)

Projective vs. inductive topology

I'm quite sure that the final topology is also called "inductive topology" and the initial topology is the "projective topology", thus there is a serious mistake in the first sentence (and in the first sentence of the article about the initial topology). This would also be consistent with "inductive topology" redirecting to "final topology" and "projective topology" redirecting to "initial topology" - so either the redirections are wrong or the articles. I think this should be fixed asap, but I haven't changed anything yet because I don't have a reference at hand. TSBM (talk) — Preceding undated comment added 12:00, 8 December 2014 (UTC)

On the given universal property

"The final topology on $X$ can be characterized by the following universal property: a function $g$ from $X$ to some space $Z$ is continuous if and only if $g\circ f_{i}$ is continuous for each i ∈ I."

Well, not really. This universal property is for the topological space $(X,\tau )$. The universal property of $\tau$ is the one given at the beginning:

$\tau$ is the finest topology such that each

$f_{i}:Y_{i}\to (X,\tau )$

is continuous.

This means explicitly: For every topology $\tau '$ on $X$ such that the $f_{i}$ are continuous, $\tau '\subseteq \tau$. (The inclusion $\tau '\subseteq \tau$ can be viewed as a morphism in the category of topologies on $X$.)

Claus from Leipzig (talk) 12:25, 17 October 2017 (UTC)

Cone versus Cocone

In the categorical description section, cones are used when cocones ( F(X) \to N ) might be desired? — Preceding unsigned comment added by 136.159.16.20 (talk) 03:57, 24 February 2018 (UTC)