Path connectedness of union of path connected spaces

In summary, the question asks if a collection of path connected subspaces with a nonempty intersection will always result in a path connected union. The answer is no, as even though elements within each subspace can be connected by a path, there is no guarantee that elements in the union outside of the intersection will be connected.
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radou
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Homework Statement



As the title suggests, Let {Aj} be a collection of path connected subspaces of some space X, and let the intersection of these subspaces be nonempty. Is U Aj path connected?

The Attempt at a Solution



Again, my answer would be no, in general.

But, since their intersection is nonempty, that means that all these subspaces share at least one element in common. So, if we choose any x from U Aj, and any y from the intersection, we can find a path which connects them, since x is in Aj, for some j, and y is in Aj, too.
 
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  • #2
However, if we now choose a point z from U Aj, that is not in the intersection of all the Aj's, then there is no guarantee that there exists a path which connects x and z. Therefore, in general, U Aj is not path connected.
 

What is path connectedness?

Path connectedness is a property of a topological space that means every pair of points in the space can be connected by a continuous path. This means that there is a continuous function that maps the interval [0,1] onto the space, where the starting point is mapped to one point and the ending point is mapped to the other point.

What does it mean for a union of path connected spaces to be path connected?

If a union of path connected spaces is path connected, it means that any two points in the union can be connected by a path that lies entirely within the union. This is true even if the two points lie in different path connected spaces within the union.

Is the union of two path connected spaces always path connected?

No, the union of two path connected spaces is not always path connected. For example, if the two spaces are disjoint from each other, there is no continuous path that can connect a point in one space to a point in the other space.

Can a union of path connected spaces be path connected if one of the spaces is not path connected?

No, if even one of the spaces in the union is not path connected, then the whole union cannot be path connected. This is because the path connectedness of the union is dependent on the path connectedness of all the individual spaces within the union.

What is an example of a union of path connected spaces that is not path connected?

An example of a union of path connected spaces that is not path connected is the union of the unit circle and the line segment [0,1] in the plane. Both of these spaces are individually path connected, but there is no continuous path that can connect a point on the unit circle to a point on the line segment.

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