Is Being Path-Connected the Key to Understanding Simply Connected Regions?

[Aldnoahz]

Hi. I am studying Multivariable Calculus and found simply connected regions difficult to understand. Why is an xy plane with the positive x-axis removed a simply connected region while an xy plane with the entire x-axis removed is not simply connected?

In the latter case, as x-axis is not defined, we shouldn't even be able to draw any curve crossing axis... So this leaves me with two separated planes in which everywhere is defined and differentiable. Then why is it still not simply connected?

How about the former case?
I am confused.

 

[andrewkirk]

I think (without looking it up to check) it's because a simply connected space is defined as one that is both connected and has the property of being able to reduce any loop to a point. The plane with the x-axis removed is not connected and hence not simply connected.

If it weren't for the 'and' the plane with an axis removed would not be simply connected.

 

[lavinia]

I think (without looking it up to check) it's because a simply connected space is defined as one that is both connected and has the property of being able to reduce any loop to a point. The plane with the x-axis removed is not connected and hence not simply connected.

If it weren't for the 'and' the plane with an axis removed would not be simply connected.

Technically I think a simply connected space is required to be path connected.

 

[Svein]

"Yet another formulation is often used in complex analysis: an open subset X of C is simply-connected if and only if both X and its complement in the Riemann sphere are connected" (https://en.wikipedia.org/wiki/Simply_connected_space).

 

[FactChecker]

In the latter case, as x-axis is not defined, we shouldn't even be able to draw any curve crossing axis... So this leaves me with two separated planes in which everywhere is defined and differentiable. Then why is it still not simply connected?

You answered it yourself. It is in two separate parts that are not connected. So it is not simply connected.

 

[andrewkirk]

You answered it yourself. It is in two separate parts that are not connected. So it is not simply connected.

That is only clear when one knows that being path-connected is an additional requirement for a space to be simply connected, as per the above discussion. Forgetting that additional criterion is very understandable since nearly all the emphasis on simple connectedness (at least in my texts) is on the ability to contract a loop.

It's also worth noting that the presence of the word 'connected' in 'simply connected' does not provide an etymological clue that connectedness is also required, as the term 'locally connected' demonstrates. Neither of local connectedness nor connectedness entails the other.

 

[FactChecker]

That is only clear when one knows that being path-connected is an additional requirement for a space to be simply connected, as per the above discussion. Forgetting that additional criterion is very understandable since nearly all the emphasis on simple connectedness (at least in my texts) is on the ability to contract a loop.

It's also worth noting that the presence of the word 'connected' in 'simply connected' does not provide an etymological clue that connectedness is also required, as the term 'locally connected' demonstrates. Neither of local connectedness nor connectedness entails the other.

I agree. There may easily be some confusion about the term. I just meant to say that it is much simpler than the OP was expecting. He had, in fact, answered the question in his own question.