Show that the set of all inner points of a set like $A$ ( a subset of real numbers) is open.
$begingroup$
My attempt:
We have to show that if $x$ is an inner point for A then it is an inner point of inner points of $A$.
There exists some positive $r$ for which the interval
$(x-r, x+r)$ is in $A$.
hint:we know that $(x-r, x+r)$ is an open set.
How should I continue?
real-analysis
edited Dec 16 '18 at 19:00
t.ysn
1397
asked Dec 16 '18 at 18:46
Arman_jrArman_jr
235
$endgroup$
add a comment |
$begingroup$
My attempt:
We have to show that if $x$ is an inner point for A then it is an inner point of inner points of $A$.
There exists some positive $r$ for which the interval
$(x-r, x+r)$ is in $A$.
hint:we know that $(x-r, x+r)$ is an open set.
How should I continue?
real-analysis
edited Dec 16 '18 at 19:00
t.ysn
1397
asked Dec 16 '18 at 18:46
Arman_jrArman_jr
235
$endgroup$
add a comment |
$begingroup$
My attempt:
We have to show that if $x$ is an inner point for A then it is an inner point of inner points of $A$.
There exists some positive $r$ for which the interval
$(x-r, x+r)$ is in $A$.
hint:we know that $(x-r, x+r)$ is an open set.
How should I continue?
real-analysis
edited Dec 16 '18 at 19:00
t.ysn
1397
asked Dec 16 '18 at 18:46
Arman_jrArman_jr
235
$endgroup$
My attempt:
We have to show that if $x$ is an inner point for A then it is an inner point of inner points of $A$.
There exists some positive $r$ for which the interval
$(x-r, x+r)$ is in $A$.
hint:we know that $(x-r, x+r)$ is an open set.
How should I continue?
real-analysis
real-analysis
edited Dec 16 '18 at 19:00
t.ysn
1397
asked Dec 16 '18 at 18:46
Arman_jrArman_jr
235
edited Dec 16 '18 at 19:00
t.ysn
1397
asked Dec 16 '18 at 18:46
Arman_jrArman_jr
235
edited Dec 16 '18 at 19:00
t.ysn
1397
edited Dec 16 '18 at 19:00
t.ysn
1397
edited Dec 16 '18 at 19:00
t.ysn
1397
1397
asked Dec 16 '18 at 18:46
Arman_jrArman_jr
235
asked Dec 16 '18 at 18:46
Arman_jrArman_jr
235
asked Dec 16 '18 at 18:46
Arman_jrArman_jr
235
235
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
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Hint: Since $(x-r,x+r)$ is open, what do we know by definition is the case for every point within $(x-r,x+r)$?
answered Dec 16 '18 at 18:48
Will FisherWill Fisher
4,05311132
$endgroup$
$begingroup$
we know that for any point like y within it there exists some r which (y-r, y+r) is in (x-r, x+r)
$endgroup$
– Arman_jr
Dec 16 '18 at 19:20
$begingroup$
Exactly, so every point of $(x-r,x+r)$ is an interior point of $(x-r,x+r)$, but $(x-r,x+r)subset A$, so every point is also an interior point of...
$endgroup$
– Will Fisher
Dec 16 '18 at 19:43
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
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votes
$begingroup$
Hint: Since $(x-r,x+r)$ is open, what do we know by definition is the case for every point within $(x-r,x+r)$?
answered Dec 16 '18 at 18:48
Will FisherWill Fisher
4,05311132
$endgroup$
$begingroup$
we know that for any point like y within it there exists some r which (y-r, y+r) is in (x-r, x+r)
$endgroup$
– Arman_jr
Dec 16 '18 at 19:20
$begingroup$
Exactly, so every point of $(x-r,x+r)$ is an interior point of $(x-r,x+r)$, but $(x-r,x+r)subset A$, so every point is also an interior point of...
$endgroup$
– Will Fisher
Dec 16 '18 at 19:43
add a comment |
$begingroup$
Hint: Since $(x-r,x+r)$ is open, what do we know by definition is the case for every point within $(x-r,x+r)$?
answered Dec 16 '18 at 18:48
Will FisherWill Fisher
4,05311132
$endgroup$
$begingroup$
we know that for any point like y within it there exists some r which (y-r, y+r) is in (x-r, x+r)
$endgroup$
– Arman_jr
Dec 16 '18 at 19:20
$begingroup$
Exactly, so every point of $(x-r,x+r)$ is an interior point of $(x-r,x+r)$, but $(x-r,x+r)subset A$, so every point is also an interior point of...
$endgroup$
– Will Fisher
Dec 16 '18 at 19:43
add a comment |
$begingroup$
Hint: Since $(x-r,x+r)$ is open, what do we know by definition is the case for every point within $(x-r,x+r)$?
answered Dec 16 '18 at 18:48
Will FisherWill Fisher
4,05311132
$endgroup$
Hint: Since $(x-r,x+r)$ is open, what do we know by definition is the case for every point within $(x-r,x+r)$?
answered Dec 16 '18 at 18:48
Will FisherWill Fisher
4,05311132
answered Dec 16 '18 at 18:48
Will FisherWill Fisher
4,05311132
answered Dec 16 '18 at 18:48
Will FisherWill Fisher
4,05311132
answered Dec 16 '18 at 18:48
Will FisherWill Fisher
4,05311132
4,05311132
$begingroup$
we know that for any point like y within it there exists some r which (y-r, y+r) is in (x-r, x+r)
$endgroup$
– Arman_jr
Dec 16 '18 at 19:20
$begingroup$
Exactly, so every point of $(x-r,x+r)$ is an interior point of $(x-r,x+r)$, but $(x-r,x+r)subset A$, so every point is also an interior point of...
$endgroup$
– Will Fisher
Dec 16 '18 at 19:43
add a comment |
$begingroup$
we know that for any point like y within it there exists some r which (y-r, y+r) is in (x-r, x+r)
$endgroup$
– Arman_jr
Dec 16 '18 at 19:20
$begingroup$
Exactly, so every point of $(x-r,x+r)$ is an interior point of $(x-r,x+r)$, but $(x-r,x+r)subset A$, so every point is also an interior point of...
$endgroup$
– Will Fisher
Dec 16 '18 at 19:43
$begingroup$
we know that for any point like y within it there exists some r which (y-r, y+r) is in (x-r, x+r)
$endgroup$
– Arman_jr
Dec 16 '18 at 19:20
$begingroup$
we know that for any point like y within it there exists some r which (y-r, y+r) is in (x-r, x+r)
$endgroup$
– Arman_jr
Dec 16 '18 at 19:20
$begingroup$
Exactly, so every point of $(x-r,x+r)$ is an interior point of $(x-r,x+r)$, but $(x-r,x+r)subset A$, so every point is also an interior point of...
$endgroup$
– Will Fisher
Dec 16 '18 at 19:43
$begingroup$
Exactly, so every point of $(x-r,x+r)$ is an interior point of $(x-r,x+r)$, but $(x-r,x+r)subset A$, so every point is also an interior point of...
$endgroup$
– Will Fisher
Dec 16 '18 at 19:43
add a comment |
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