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Regarding "...a function f : X → Y is continuous...if the preimage of every closed set of Y is closed in X." at the end of the second paragraph:
Forgive me if I'm mistaken, but in general, preimages of closed sets being closed does not ensure continuity. E.g. for f: [0,1] -> (0,1] where f(x) = x (for x != 0) and f(0) = 1/2, f is not continuous. Perhaps something is assumed that I missed? 203.150.100.189 08:38, 20 March 2007 (UTC)
Unless I'm going crazy, "An open map is also closed if and only if it is surjective" is definitely not true. Just include X into two (disjoint) copies of itself. —Preceding unsigned comment added by 24.19.0.156 ( talk) 00:54, 8 February 2011 (UTC)
I was wondering if isometries in metric spaces are open. — Preceding unsigned comment added by Noix07 ( talk • contribs) 15:35, 16 December 2013 (UTC)
A sentence in the "Examples" section says that the floor function is open. This seems doubtful; the image of open interval (1/3, 2/3) under this map is the singleton set {0}, which is not open. (Right?) If I'm right, would someone please correct that line?
Norbornene ( talk) 20:29, 3 January 2017 (UTC)
As the definition is currently written, a relatively open map seems like it's the same thing as an open map. (This makes the crazy capital letters "WARNING" a little hard to understand.)
My guess is that it should say is the image of *considered as a topological space under the subspace topology*, which would make sense and would make the two definitions different. However, I can't currently check the reference to be sure. Is this correct?
Nathaniel Virgo ( talk) 07:09, 27 October 2020 (UTC)
The article says:
"3. A map is called an open map or a strongly open map if it satisfies any of the following equivalent conditions: ... For every and every neighborhood of (however small), is a neighborhood of .
This is how I understand the above: By replacing one or two of the occurrences of "neighborhood" with "open neighborhood" in the above statement we can get 4 different statements:
A) For every and every neighborhood of , is a neighborhood of .
B) For every and every open neighborhood of , is a neighborhood of .
C) For every and every neighborhood of , is an open neighborhood of .
D) For every and every open neighborhood of , is an open neighborhood of .
Indeed A), B) and D) are equivalent to the main definition of (strongly) open map, which is that any open subset of is mapped to an open subset of . However C) is definitely not equivalent to being an open map. As a counterexample take the real line endowed with the Euclidean topology and the identity function . Obviously is an open map, but C) does not hold for . For example is a neighborhood of but maps that interval to itself and it is not an open neighborhood of .
So I would suggest removing the sentence "* Either instance of the word "neighborhood" in this statement can be replaced with "open neighborhood" and the resulting statement would still characterize strongly open maps." and just listing conditions A), B) and D) explicitly and saying that they are equivalent to the main definition of (strongly) open map. Vesselin.atanasov ( talk) 03:14, 10 March 2023 (UTC)
![]() | This article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
|
Regarding "...a function f : X → Y is continuous...if the preimage of every closed set of Y is closed in X." at the end of the second paragraph:
Forgive me if I'm mistaken, but in general, preimages of closed sets being closed does not ensure continuity. E.g. for f: [0,1] -> (0,1] where f(x) = x (for x != 0) and f(0) = 1/2, f is not continuous. Perhaps something is assumed that I missed? 203.150.100.189 08:38, 20 March 2007 (UTC)
Unless I'm going crazy, "An open map is also closed if and only if it is surjective" is definitely not true. Just include X into two (disjoint) copies of itself. —Preceding unsigned comment added by 24.19.0.156 ( talk) 00:54, 8 February 2011 (UTC)
I was wondering if isometries in metric spaces are open. — Preceding unsigned comment added by Noix07 ( talk • contribs) 15:35, 16 December 2013 (UTC)
A sentence in the "Examples" section says that the floor function is open. This seems doubtful; the image of open interval (1/3, 2/3) under this map is the singleton set {0}, which is not open. (Right?) If I'm right, would someone please correct that line?
Norbornene ( talk) 20:29, 3 January 2017 (UTC)
As the definition is currently written, a relatively open map seems like it's the same thing as an open map. (This makes the crazy capital letters "WARNING" a little hard to understand.)
My guess is that it should say is the image of *considered as a topological space under the subspace topology*, which would make sense and would make the two definitions different. However, I can't currently check the reference to be sure. Is this correct?
Nathaniel Virgo ( talk) 07:09, 27 October 2020 (UTC)
The article says:
"3. A map is called an open map or a strongly open map if it satisfies any of the following equivalent conditions: ... For every and every neighborhood of (however small), is a neighborhood of .
This is how I understand the above: By replacing one or two of the occurrences of "neighborhood" with "open neighborhood" in the above statement we can get 4 different statements:
A) For every and every neighborhood of , is a neighborhood of .
B) For every and every open neighborhood of , is a neighborhood of .
C) For every and every neighborhood of , is an open neighborhood of .
D) For every and every open neighborhood of , is an open neighborhood of .
Indeed A), B) and D) are equivalent to the main definition of (strongly) open map, which is that any open subset of is mapped to an open subset of . However C) is definitely not equivalent to being an open map. As a counterexample take the real line endowed with the Euclidean topology and the identity function . Obviously is an open map, but C) does not hold for . For example is a neighborhood of but maps that interval to itself and it is not an open neighborhood of .
So I would suggest removing the sentence "* Either instance of the word "neighborhood" in this statement can be replaced with "open neighborhood" and the resulting statement would still characterize strongly open maps." and just listing conditions A), B) and D) explicitly and saying that they are equivalent to the main definition of (strongly) open map. Vesselin.atanasov ( talk) 03:14, 10 March 2023 (UTC)