(1) Local connectedness is preserved by quotient maps. Argued by Henno Brandsma on MSE. Also Proposition 12 in Section 11.6 on page 112 of Bourbaki's general topology book.
(2) 27F.1 of Willard says,
- The continuous image of a locally connected space need not be locally connected.
(3) Local path connectedness is preserved by quotient maps. Henno Brandsma's post adapts directly, but this is also explicitly stated by Braza in Lemma 3.4 of this article.
(4) I didn't find an example of local path connectedness (or local connectedness) not being preserved by images. Maybe relevant: If a non-locally path connected image of the interval $[0, 1]$ exists, then it is non-Hausdorff because proper maps to Hausdorff spaces are closed maps. See Hahn–Mazurkiewicz theorem; also see Willard for Peano Space.
(5) For some applications to traits:
(1) We could improve the trait S196|P41, by using local connectedness of the closed long ray (which is derived). This trait is currently derived already. Also, the reference in the trait file not really adding a lot to this specific trait.
(2) We could add a secondary argument to S139|P42 which concludes immediately. We could also replace the whole text just with the application of this fact (though the current text seems fine). We could also just not do anything.
So I propose we add the following standard phrasing to the meta-property list of these traits. We should also discuss if it's worth adding a reference / link at the end in parentheses.
This property is preserved by quotient maps.
(1) Local connectedness is preserved by quotient maps. Argued by Henno Brandsma on MSE. Also Proposition 12 in Section 11.6 on page 112 of Bourbaki's general topology book.
(2) 27F.1 of Willard says,
(3) Local path connectedness is preserved by quotient maps. Henno Brandsma's post adapts directly, but this is also explicitly stated by Braza in Lemma 3.4 of this article.
(4) I didn't find an example of local path connectedness (or local connectedness) not being preserved by images. Maybe relevant: If a non-locally path connected image of the interval$[0, 1]$ exists, then it is non-Hausdorff because proper maps to Hausdorff spaces are closed maps. See Hahn–Mazurkiewicz theorem; also see Willard for Peano Space.
(5) For some applications to traits:
(1) We could improve the trait S196|P41, by using local connectedness of the closed long ray (which is derived). This trait is currently derived already. Also, the reference in the trait file not really adding a lot to this specific trait.
(2) We could add a secondary argument to S139|P42 which concludes immediately. We could also replace the whole text just with the application of this fact (though the current text seems fine). We could also just not do anything.
So I propose we add the following standard phrasing to the meta-property list of these traits. We should also discuss if it's worth adding a reference / link at the end in parentheses.