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Simply connected

dwsmith

Well-known member
Feb 1, 2012
1,673
A set $S$ is called star shaped if there exist a point $z_0\in S$ such that the line segment between $z_0$ and any point in $S$ is contained in $S$. Prove that a star shaped set is simply connected.


Let $\gamma_1$ and $\gamma_2$ be two closed curves with the same initial and end points $z_0\in S$ which are both parameterized on $I$.
Define
$$
F:I^2\to S \quad \text{by} \quad
F(s,t) = (1 - t)\gamma_1(s) + t\gamma_2(s),
$$
where $I = [0,1]$.
Then $F(s,t)$ is uniformly continuous since $I^2$ is compact and $F(s,t)$ is the sum of continuous functions.
So $F(s,0) = \gamma_1(s)$, $F(s,1) = \gamma_2(s)$, $F(0,t) = z_0$, and $F(1,t) = z_0$.
Additionally, any closed $\gamma_i$ is homotopic to a point in $S$.
So $\int_{\gamma_i}F = 0$.
Therefore $S$ is simply connected.

Correct?
 
Last edited:

CaptainBlack

Well-known member
Jan 26, 2012
890
A set $S$ is called star shaped if there exist a point $z_0\in S$ such that the line segment between $z_0$ and any point in $S$ is contained in $S$. Prove that a star shaped set is simply connected.


Let $\gamma_1$ and $\gamma_2$ be two closed curves with the same initial and end points $z_0\in S$ which are both parameterized on $I$.
Define
$$
F:I^2\to S \quad \text{by} \quad
F(s,t) = (1 - t)\gamma_1(s) + t\gamma_2(s),
$$
where $I = [0,1]$.
Then $F(s,t)$ is uniformly continuous since $I^2$ is compact and $F(s,t)$ is the sum of continuous functions.
So $F(s,0) = \gamma_1(s)$, $F(s,1) = \gamma_2(s)$, $F(0,t) = z_0$, and $F(1,t) = z_0$.
Additionally, any closed $\gamma_i$ is homotopic to a point in $S$.
So $\int_{\gamma_i}F = 0$.
Therefore $S$ is simply connected.

Correct?
Where did you use the fact that S is star shaped?

CB
 

dwsmith

Well-known member
Feb 1, 2012
1,673
Where did you use the fact that S is star shaped?

CB
Then the first line should say let S be star shaped, i.e. S is path connected. Then is it fine?
 

CaptainBlack

Well-known member
Jan 26, 2012
890
Then the first line should say let S be star shaped, i.e. S is path connected. Then is it fine?
But then where do you use the property.

(that it is star shaped should indicate that you can connect any two points by a path consisting of a pair of line segments in \(S\) connecting the two points via \( z_0 \) )

CB
 

dwsmith

Well-known member
Feb 1, 2012
1,673
But then where do you use the property.

(that it is star shaped should indicate that you can connect any two points by a path consisting of a pair of line segments in \(S\) connecting the two points via \( z_0 \) )

CB
Do I need to stated the definition of star shaped? The fact that it is star shaped we know any points in S can be connected by a path contained in S.
 

CaptainBlack

Well-known member
Jan 26, 2012
890
Do I need to stated the definition of star shaped? The fact that it is star shaped we know any points in S can be connected by a path contained in S.
I would suggest you explictly give the path consisting of the two segments, noting that the star connectedness guarantees this path is in \( S\) . That proves it is connected, which leaves the simply part...

Cb
 
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dwsmith

Well-known member
Feb 1, 2012
1,673
I would suggest you explictly give the path consisting of the two segments, noting that the star connectedness guarantees this path is in \( S\) .

Cb
Do you mean by stating the gamma_1 is closed loop starting and ending at z_0 and gamma_2 is just a point curve at z_0?
 

CaptainBlack

Well-known member
Jan 26, 2012
890
Do you mean by stating the gamma_1 is closed loop starting and ending at z_0 and gamma_2 is just a point curve at z_0?
Ignore my last post, the question remains, where do you use the star property, and is your \( z_0 \) the special \( z_0\) of the star definition?

Now I think about it you haven't specified what type of space we are dealing with, by the look of it \( S \) needs to be a subset of vector space.
 
Last edited:

dwsmith

Well-known member
Feb 1, 2012
1,673
Ignore my last post, the question remains, where do you use the star property, and is your \( z_0 \) the special \( z_0\) of the star definition?

Now I think about it you haven't specified what type of space we are dealing with, by the look of it \( S \) needs to be a subset of vector space.
S is in C and my z_0 is the z_0 from the definition.
 

dwsmith

Well-known member
Feb 1, 2012
1,673
How can I show that the image of F lies in S?

How can I show that it is okay $\gamma_1$ to start at $z_0$?