Proving Path Connectivity of Set S with Rational Line Segments

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In summary: Since all the points are rational, you can always find a point on a line segment between two points that is also rational. This means you can continuously move from a to (0,1) to b, showing that S is path connected.
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muzak
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Homework Statement


I am given a set S consisting of the union of line segments from the point (0,1) to points (x,0) x-values are rationals from [0,1]. I want to show that this is path connected.


Homework Equations


Finding a continuous function f:[0,1] -> S such that f(0) = a and f(1) = b where a,b are in the set.


The Attempt at a Solution



I don't know how to show this part, do I attempt to show continuity in the topological sense? I don't even know how to attempt that. I'm having a hard time conceptualizing continuity with this.
 
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  • #2
muzak said:

Homework Statement


I am given a set S consisting of the union of line segments from the point (0,1) to points (x,0) x-values are rationals from [0,1]. I want to show that this is path connected.

Homework Equations


Finding a continuous function f:[0,1] -> S such that f(0) = a and f(1) = b where a,b are in the set.

The Attempt at a Solution



I don't know how to show this part, do I attempt to show continuity in the topological sense? I don't even know how to attempt that. I'm having a hard time conceptualizing continuity with this.

You've got a bunch of line segments that are all connected to (0,1). Just go from a to (0,1) and then from there to b.
 

1. What is the definition of path connected in mathematics?

Path connectedness is a topological property that describes a space in which any two points can be connected by a continuous path. In simpler terms, it means that there are no holes or gaps in the space and it is possible to go from one point to another without leaving the space.

2. How is path connectedness different from connectedness?

Connectedness refers to a space in which it is possible to find a continuous path between any two points. However, path connectedness takes it a step further by requiring that the path itself also lies within the space. In other words, path connectedness implies connectedness, but not vice versa.

3. Why is path connectedness an important concept in mathematics?

Path connectedness is important because it allows us to analyze the continuity and connectedness of a space. It also helps in understanding the behavior of functions and mappings on that space. Additionally, many important topological properties, such as compactness and completeness, are defined in terms of path connectedness.

4. Can a disconnected space be path connected?

No, a disconnected space cannot be path connected. This is because a disconnected space can be divided into two or more separate pieces, and therefore, there is no way to find a continuous path between points in different pieces. However, it is possible for a connected space to not be path connected.

5. How can I prove that a space is path connected?

To prove that a space is path connected, you can show that any two points in the space can be connected by a continuous path. This can be done by explicitly constructing a path between the two points or by showing that the space satisfies the definition of path connectedness. You can also use the fact that path connectedness is a topological property and use known theorems or techniques to prove it.

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