Set Point Topology-product and connected spaces.

[TheForumLord]

Homework Statement

1. Prove that the space C([0,1]) is arc-connected (C([0,1]) = real continuous functions onto [0,1] with the metric max|f(x)-g(x)| )

2. Prove that in a product space of infinite many spaces, such as in each space there is more than one point, every point is an accumulation point.

3. Prove that $$R_{CF}$$ is path-connected but not arc-connected.

Homework Equations

The Attempt at a Solution

I'm really bad at all the path-connected&arc-connected subject so I can't really understand how I should start solving these 3 problems...I have no clue about them...


Tnx in advance

 

[mighty2000]

for any help!

Dear student,

I will provide some guidance on how to approach these problems, but it is important that you try to solve them on your own first. It is also helpful to review the definitions of arc-connected, path-connected, and accumulation points before attempting the proofs.

1. To prove that a space is arc-connected, you need to show that there exists a continuous function that maps any two points in the space to a continuous curve between them. In this case, the space C([0,1]) consists of real continuous functions, so you need to construct a continuous function that maps two functions f and g in C([0,1]) to a continuous curve that connects them. To do this, you can consider the linear interpolation between f and g, which is a continuous function that connects the two points.

2. In a product space of infinite many spaces, you can use the definition of an accumulation point to show that every point is an accumulation point. An accumulation point of a set A is a point x such that for any neighborhood U of x, there exists a point y in U that is also in A. In the product space, you can construct a neighborhood for each point that contains infinitely many points from each space, which means that each point will have infinitely many points in its neighborhood and therefore is an accumulation point.

3. To prove that R_{CF} is path-connected but not arc-connected, you need to show that there exists a continuous function that maps any two points in R_{CF} to a continuous curve between them, but there does not exist a continuous function that maps any two points to an arc. To do this, you can consider the definition of R_{CF} and how it differs from a typical arc-connected space. In particular, R_{CF} is defined as the set of all real continuous functions on [0,1] that are equal to 0 at every rational point and equal to 1 at every irrational point. This means that any continuous curve between two points in R_{CF} must pass through both rational and irrational points, which is not possible for an arc.