Questions about the proof sketch of “${x}$ is a deformation retract of $overline{St}(x)$”
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Munkres Topology Section 83
First question: Is "obvious deformation" the straight line homotopy
$F(b,t) = overline{St}(x) times I to overline{St}(x)$?
Second question: Is the "result" in "This result follows" making a reference to a previous exercise? If not, then what?
Third question: Is the "general result" in "but the general result requires one" referring to "its restriction to each subspace $A_{alpha} times I$ is continuous"? If not, then what?
I am not asking for proofs but only that to understand this sketch that Munkres has given.
general-topology graph-theory algebraic-topology deformation-theory retraction
asked Nov 16 at 3:17
Jack Bauer
1,236531
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Munkres Topology Section 83
First question: Is "obvious deformation" the straight line homotopy
$F(b,t) = overline{St}(x) times I to overline{St}(x)$?
Second question: Is the "result" in "This result follows" making a reference to a previous exercise? If not, then what?
Third question: Is the "general result" in "but the general result requires one" referring to "its restriction to each subspace $A_{alpha} times I$ is continuous"? If not, then what?
I am not asking for proofs but only that to understand this sketch that Munkres has given.
general-topology graph-theory algebraic-topology deformation-theory retraction
asked Nov 16 at 3:17
Jack Bauer
1,236531
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Munkres Topology Section 83
First question: Is "obvious deformation" the straight line homotopy
$F(b,t) = overline{St}(x) times I to overline{St}(x)$?
Second question: Is the "result" in "This result follows" making a reference to a previous exercise? If not, then what?
Third question: Is the "general result" in "but the general result requires one" referring to "its restriction to each subspace $A_{alpha} times I$ is continuous"? If not, then what?
I am not asking for proofs but only that to understand this sketch that Munkres has given.
general-topology graph-theory algebraic-topology deformation-theory retraction
asked Nov 16 at 3:17
Jack Bauer
1,236531
Munkres Topology Section 83
First question: Is "obvious deformation" the straight line homotopy
$F(b,t) = overline{St}(x) times I to overline{St}(x)$?
Second question: Is the "result" in "This result follows" making a reference to a previous exercise? If not, then what?
Third question: Is the "general result" in "but the general result requires one" referring to "its restriction to each subspace $A_{alpha} times I$ is continuous"? If not, then what?
I am not asking for proofs but only that to understand this sketch that Munkres has given.
general-topology graph-theory algebraic-topology deformation-theory retraction
general-topology graph-theory algebraic-topology deformation-theory retraction
asked Nov 16 at 3:17
Jack Bauer
1,236531
asked Nov 16 at 3:17
Jack Bauer
1,236531
asked Nov 16 at 3:17
Jack Bauer
1,236531
asked Nov 16 at 3:17
Jack Bauer
1,236531
asked Nov 16 at 3:17
Jack Bauer
1,236531
1,236531
add a comment |
add a comment |
1 Answer
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First question: Is "obvious deformation" the straight line homotopy
Yes, loosely speaking if $b$ belongs to arc $A$, then $$F(b, t)=f^{-1}big(tf(b) + (1-t)f(x)big)$$ where $f:Ato[0,1]$ is a homeomorphism. Care must be taken though, for example we have to pick homeomorphisms $f$ in such a way that $f(x)$ is always say $1$.
Second question: Is the "result" in "This result follows" making a reference to a previous exercise? If not, then what?
The "result" word refers to "the fact that a map $F$ is continuous if its restriction to each subspace (...) is continuous".
The "follows from (...) in case $overline{St}(x)$ is a union of finite number of arcs" refers to the Pasting Lemma which is essentially what your exercise $9$ says.
Third question: Is the "general result" in "but the general result requires one" referring to "its restriction to each subspace $A_{alpha} times I$ is continuous"? If not, then what?
The general result refers to the same statement (that $F$ is continuous if its restrictions are) but in the case when $overline{St}(x)$ is not a union of finite number of arc. For that you need the fact that the topology on $overline{St}(x)times I$ is coherent with ${A_alphatimes I}$.
answered Nov 16 at 10:47
freakish
10.4k1526
I cannot believe that was not ambiguous. Thank you, freakish
– Jack Bauer
Nov 16 at 14:00
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
First question: Is "obvious deformation" the straight line homotopy
Yes, loosely speaking if $b$ belongs to arc $A$, then $$F(b, t)=f^{-1}big(tf(b) + (1-t)f(x)big)$$ where $f:Ato[0,1]$ is a homeomorphism. Care must be taken though, for example we have to pick homeomorphisms $f$ in such a way that $f(x)$ is always say $1$.
Second question: Is the "result" in "This result follows" making a reference to a previous exercise? If not, then what?
The "result" word refers to "the fact that a map $F$ is continuous if its restriction to each subspace (...) is continuous".
The "follows from (...) in case $overline{St}(x)$ is a union of finite number of arcs" refers to the Pasting Lemma which is essentially what your exercise $9$ says.
Third question: Is the "general result" in "but the general result requires one" referring to "its restriction to each subspace $A_{alpha} times I$ is continuous"? If not, then what?
The general result refers to the same statement (that $F$ is continuous if its restrictions are) but in the case when $overline{St}(x)$ is not a union of finite number of arc. For that you need the fact that the topology on $overline{St}(x)times I$ is coherent with ${A_alphatimes I}$.
answered Nov 16 at 10:47
freakish
10.4k1526
I cannot believe that was not ambiguous. Thank you, freakish
– Jack Bauer
Nov 16 at 14:00
add a comment |
up vote
1
down vote
accepted
First question: Is "obvious deformation" the straight line homotopy
Yes, loosely speaking if $b$ belongs to arc $A$, then $$F(b, t)=f^{-1}big(tf(b) + (1-t)f(x)big)$$ where $f:Ato[0,1]$ is a homeomorphism. Care must be taken though, for example we have to pick homeomorphisms $f$ in such a way that $f(x)$ is always say $1$.
Second question: Is the "result" in "This result follows" making a reference to a previous exercise? If not, then what?
The "result" word refers to "the fact that a map $F$ is continuous if its restriction to each subspace (...) is continuous".
The "follows from (...) in case $overline{St}(x)$ is a union of finite number of arcs" refers to the Pasting Lemma which is essentially what your exercise $9$ says.
Third question: Is the "general result" in "but the general result requires one" referring to "its restriction to each subspace $A_{alpha} times I$ is continuous"? If not, then what?
The general result refers to the same statement (that $F$ is continuous if its restrictions are) but in the case when $overline{St}(x)$ is not a union of finite number of arc. For that you need the fact that the topology on $overline{St}(x)times I$ is coherent with ${A_alphatimes I}$.
answered Nov 16 at 10:47
freakish
10.4k1526
I cannot believe that was not ambiguous. Thank you, freakish
– Jack Bauer
Nov 16 at 14:00
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
First question: Is "obvious deformation" the straight line homotopy
Yes, loosely speaking if $b$ belongs to arc $A$, then $$F(b, t)=f^{-1}big(tf(b) + (1-t)f(x)big)$$ where $f:Ato[0,1]$ is a homeomorphism. Care must be taken though, for example we have to pick homeomorphisms $f$ in such a way that $f(x)$ is always say $1$.
Second question: Is the "result" in "This result follows" making a reference to a previous exercise? If not, then what?
The "result" word refers to "the fact that a map $F$ is continuous if its restriction to each subspace (...) is continuous".
The "follows from (...) in case $overline{St}(x)$ is a union of finite number of arcs" refers to the Pasting Lemma which is essentially what your exercise $9$ says.
Third question: Is the "general result" in "but the general result requires one" referring to "its restriction to each subspace $A_{alpha} times I$ is continuous"? If not, then what?
The general result refers to the same statement (that $F$ is continuous if its restrictions are) but in the case when $overline{St}(x)$ is not a union of finite number of arc. For that you need the fact that the topology on $overline{St}(x)times I$ is coherent with ${A_alphatimes I}$.
answered Nov 16 at 10:47
freakish
10.4k1526
First question: Is "obvious deformation" the straight line homotopy
Yes, loosely speaking if $b$ belongs to arc $A$, then $$F(b, t)=f^{-1}big(tf(b) + (1-t)f(x)big)$$ where $f:Ato[0,1]$ is a homeomorphism. Care must be taken though, for example we have to pick homeomorphisms $f$ in such a way that $f(x)$ is always say $1$.
Second question: Is the "result" in "This result follows" making a reference to a previous exercise? If not, then what?
The "result" word refers to "the fact that a map $F$ is continuous if its restriction to each subspace (...) is continuous".
The "follows from (...) in case $overline{St}(x)$ is a union of finite number of arcs" refers to the Pasting Lemma which is essentially what your exercise $9$ says.
Third question: Is the "general result" in "but the general result requires one" referring to "its restriction to each subspace $A_{alpha} times I$ is continuous"? If not, then what?
The general result refers to the same statement (that $F$ is continuous if its restrictions are) but in the case when $overline{St}(x)$ is not a union of finite number of arc. For that you need the fact that the topology on $overline{St}(x)times I$ is coherent with ${A_alphatimes I}$.
answered Nov 16 at 10:47
freakish
10.4k1526
answered Nov 16 at 10:47
freakish
10.4k1526
answered Nov 16 at 10:47
freakish
10.4k1526
answered Nov 16 at 10:47
freakish
10.4k1526
10.4k1526
I cannot believe that was not ambiguous. Thank you, freakish
– Jack Bauer
Nov 16 at 14:00
add a comment |
I cannot believe that was not ambiguous. Thank you, freakish
– Jack Bauer
Nov 16 at 14:00
I cannot believe that was not ambiguous. Thank you, freakish
– Jack Bauer
Nov 16 at 14:00
I cannot believe that was not ambiguous. Thank you, freakish
– Jack Bauer
Nov 16 at 14:00
add a comment |
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