Timeline for Can a quotient space of a locally convex space have finer topology that its domain?

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Jun 23, 2020 at 8:31	comment	added	Taras Banakh		Ok, then take a metrizable countably-dimensional lcs $X_1$ and the countably dimensional lcs $X_2$ with the strongest lcs topology (it is usually called the space $\varphi)$. Then $X_2$ is the image of $X_1\times X_2$ and algebraically, $X_2$ is isomorphic to $X_1\times X_2$. The space $X_2$ has the finer topology than $X_1\times X_2$ since the topology of $X_2$ is the largest lcs topology on $X_2$.
Jun 23, 2020 at 8:17	comment	added	ABIM		@TarasBanakh Following your comment, I realized my initial formulation was trivial. I rephrased the question to better express what I had in mind.
Jun 23, 2020 at 8:16	history	edited	ABIM	CC BY-SA 4.0	added 27 characters in body; edited title
Jun 23, 2020 at 7:52	history	edited	YCor	CC BY-SA 4.0	removed capitals from title, added tag
Jun 23, 2020 at 5:09	comment	added	Taras Banakh		Yes. Take a metrizable lcs $X_1$ and a non-metrizable lcs $X_2$. Then the projection operator $X_1\times X_2\to X_1$ is not a homeomorphism.
S Jun 23, 2020 at 1:23	history	suggested	ABIM	CC BY-SA 4.0	Clarified OP's Title, added a link since OP describes a quotient map implicitly, and rephrased in terms of the quotient map (now that the terminology is explicit).
Jun 23, 2020 at 1:22	review	Suggested edits
S Jun 23, 2020 at 1:23
Jun 23, 2020 at 1:12	history	edited	ABIM	CC BY-SA 4.0	added 116 characters in body; edited tags
Jun 23, 2020 at 0:55	history	asked	ABIM	CC BY-SA 4.0