Why must a continuous function on the Long Line eventually be constant? The 2019 Stack...
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Why must a continuous function on the Long Line eventually be constant?
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$begingroup$
I've seen a couple of texts, including this answer on another question
mention that any continuous function on the Long Line must eventually be constant, but I've not seen the reasoning as to why.
Why must this be true?
general-topology functions
edited Mar 21 at 17:04
Andrews
1,2812423
asked Mar 21 at 15:38
ShufflepantsShufflepants
238111
$endgroup$
|
show 4 more comments
$begingroup$
I've seen a couple of texts, including this answer on another question
mention that any continuous function on the Long Line must eventually be constant, but I've not seen the reasoning as to why.
Why must this be true?
general-topology functions
edited Mar 21 at 17:04
Andrews
1,2812423
asked Mar 21 at 15:38
ShufflepantsShufflepants
238111
$endgroup$
$begingroup$
Have you seen the proof that a continuous (real-valued) function on the uncountable ordinal $omega_1$ is eventually constant? It's basically the same idea.
$endgroup$
– Nate Eldredge
Mar 21 at 15:45
$begingroup$
@NateEldredge No.
$endgroup$
– Shufflepants
Mar 21 at 15:49
$begingroup$
@NateEldredge A proof can be found here e.g. if you assume the pressing down lemma.
$endgroup$
– Henno Brandsma
Mar 21 at 17:13
$begingroup$
@HennoBrandsma I'm afraid I don't really understand that proof. I can't see how the pressing down lemma prevents a function such as sin(x) being defined over the entire domain, remaining continuous, and never ending up constant. And if it can't, at what point must sin(x) become constant?
$endgroup$
– Shufflepants
Mar 21 at 20:24
$begingroup$
@Shufflepants you cannot define $sin(x)$ on the long line. What is $sin(alpha, t)$ for an ordinal $alpha$ and $tin [0,1)$ in a continuous way?
$endgroup$
– Henno Brandsma
Mar 21 at 20:27
|
show 4 more comments
$begingroup$
I've seen a couple of texts, including this answer on another question
mention that any continuous function on the Long Line must eventually be constant, but I've not seen the reasoning as to why.
Why must this be true?
general-topology functions
edited Mar 21 at 17:04
Andrews
1,2812423
asked Mar 21 at 15:38
ShufflepantsShufflepants
238111
$endgroup$
I've seen a couple of texts, including this answer on another question
mention that any continuous function on the Long Line must eventually be constant, but I've not seen the reasoning as to why.
Why must this be true?
general-topology functions
general-topology functions
edited Mar 21 at 17:04
Andrews
1,2812423
asked Mar 21 at 15:38
ShufflepantsShufflepants
238111
edited Mar 21 at 17:04
Andrews
1,2812423
asked Mar 21 at 15:38
ShufflepantsShufflepants
238111
edited Mar 21 at 17:04
Andrews
1,2812423
edited Mar 21 at 17:04
Andrews
1,2812423
edited Mar 21 at 17:04
Andrews
1,2812423
1,2812423
asked Mar 21 at 15:38
ShufflepantsShufflepants
238111
asked Mar 21 at 15:38
ShufflepantsShufflepants
238111
asked Mar 21 at 15:38
ShufflepantsShufflepants
238111
238111
$begingroup$
Have you seen the proof that a continuous (real-valued) function on the uncountable ordinal $omega_1$ is eventually constant? It's basically the same idea.
$endgroup$
– Nate Eldredge
Mar 21 at 15:45
$begingroup$
@NateEldredge No.
$endgroup$
– Shufflepants
Mar 21 at 15:49
$begingroup$
@NateEldredge A proof can be found here e.g. if you assume the pressing down lemma.
$endgroup$
– Henno Brandsma
Mar 21 at 17:13
$begingroup$
@HennoBrandsma I'm afraid I don't really understand that proof. I can't see how the pressing down lemma prevents a function such as sin(x) being defined over the entire domain, remaining continuous, and never ending up constant. And if it can't, at what point must sin(x) become constant?
$endgroup$
– Shufflepants
Mar 21 at 20:24
$begingroup$
@Shufflepants you cannot define $sin(x)$ on the long line. What is $sin(alpha, t)$ for an ordinal $alpha$ and $tin [0,1)$ in a continuous way?
$endgroup$
– Henno Brandsma
Mar 21 at 20:27
|
show 4 more comments
$begingroup$
Have you seen the proof that a continuous (real-valued) function on the uncountable ordinal $omega_1$ is eventually constant? It's basically the same idea.
$endgroup$
– Nate Eldredge
Mar 21 at 15:45
$begingroup$
@NateEldredge No.
$endgroup$
– Shufflepants
Mar 21 at 15:49
$begingroup$
@NateEldredge A proof can be found here e.g. if you assume the pressing down lemma.
$endgroup$
– Henno Brandsma
Mar 21 at 17:13
$begingroup$
@HennoBrandsma I'm afraid I don't really understand that proof. I can't see how the pressing down lemma prevents a function such as sin(x) being defined over the entire domain, remaining continuous, and never ending up constant. And if it can't, at what point must sin(x) become constant?
$endgroup$
– Shufflepants
Mar 21 at 20:24
$begingroup$
@Shufflepants you cannot define $sin(x)$ on the long line. What is $sin(alpha, t)$ for an ordinal $alpha$ and $tin [0,1)$ in a continuous way?
$endgroup$
– Henno Brandsma
Mar 21 at 20:27
$begingroup$
Have you seen the proof that a continuous (real-valued) function on the uncountable ordinal $omega_1$ is eventually constant? It's basically the same idea.
$endgroup$
– Nate Eldredge
Mar 21 at 15:45
$begingroup$
Have you seen the proof that a continuous (real-valued) function on the uncountable ordinal $omega_1$ is eventually constant? It's basically the same idea.
$endgroup$
– Nate Eldredge
Mar 21 at 15:45
$begingroup$
@NateEldredge No.
$endgroup$
– Shufflepants
Mar 21 at 15:49
$begingroup$
@NateEldredge No.
$endgroup$
– Shufflepants
Mar 21 at 15:49
$begingroup$
@NateEldredge A proof can be found here e.g. if you assume the pressing down lemma.
$endgroup$
– Henno Brandsma
Mar 21 at 17:13
$begingroup$
@NateEldredge A proof can be found here e.g. if you assume the pressing down lemma.
$endgroup$
– Henno Brandsma
Mar 21 at 17:13
$begingroup$
@HennoBrandsma I'm afraid I don't really understand that proof. I can't see how the pressing down lemma prevents a function such as sin(x) being defined over the entire domain, remaining continuous, and never ending up constant. And if it can't, at what point must sin(x) become constant?
$endgroup$
– Shufflepants
Mar 21 at 20:24
$begingroup$
@HennoBrandsma I'm afraid I don't really understand that proof. I can't see how the pressing down lemma prevents a function such as sin(x) being defined over the entire domain, remaining continuous, and never ending up constant. And if it can't, at what point must sin(x) become constant?
$endgroup$
– Shufflepants
Mar 21 at 20:24
$begingroup$
@Shufflepants you cannot define $sin(x)$ on the long line. What is $sin(alpha, t)$ for an ordinal $alpha$ and $tin [0,1)$ in a continuous way?
$endgroup$
– Henno Brandsma
Mar 21 at 20:27
$begingroup$
@Shufflepants you cannot define $sin(x)$ on the long line. What is $sin(alpha, t)$ for an ordinal $alpha$ and $tin [0,1)$ in a continuous way?
$endgroup$
– Henno Brandsma
Mar 21 at 20:27
|
show 4 more comments
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$begingroup$
Have you seen the proof that a continuous (real-valued) function on the uncountable ordinal $omega_1$ is eventually constant? It's basically the same idea.
$endgroup$
– Nate Eldredge
Mar 21 at 15:45
$begingroup$
@NateEldredge No.
$endgroup$
– Shufflepants
Mar 21 at 15:49
$begingroup$
@NateEldredge A proof can be found here e.g. if you assume the pressing down lemma.
$endgroup$
– Henno Brandsma
Mar 21 at 17:13
$begingroup$
@HennoBrandsma I'm afraid I don't really understand that proof. I can't see how the pressing down lemma prevents a function such as sin(x) being defined over the entire domain, remaining continuous, and never ending up constant. And if it can't, at what point must sin(x) become constant?
$endgroup$
– Shufflepants
Mar 21 at 20:24
$begingroup$
@Shufflepants you cannot define $sin(x)$ on the long line. What is $sin(alpha, t)$ for an ordinal $alpha$ and $tin [0,1)$ in a continuous way?
$endgroup$
– Henno Brandsma
Mar 21 at 20:27