Simple Connectedness of [0,1]^k

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In summary, the conversation discusses the preservation of simple connectedness by k-cubes. It is shown that continuous maps do not necessarily preserve simple connected sets, but it is questioned whether this is true for [0,1]^k. An example is given for c(x) = (cos(2pix), sin(2pix)) for x in [0,1]. It is also mentioned that if M is a k-manifold embedded in R^n and c:[0,1]^k-->M is a k-cube homeomorphic onto its image, then the image is simply connected due to its trivial fundamental group.
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quasar987
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I am wondering if k-cubes preserve simple connectedness. I.e., is the image of [0,1]^k by a continuous function c:[0,1]^k-->R^n simply connected?

Wiki has shown me that continuous maps do not take simple connected sets to simple connected sets. For instance exp:C-->C takes C to C\{0}.

But is it true for [0,1]^k ?
 
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  • #2
What about c(x) = (cos(2pix), sin(2pix)) for x in [0,1]?
 
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Yeah. :P
 
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On the other hand, if M is a k-manifold imbedded in R^n and c:[0,1]^k-->M is a k-cube homeomorphic onto its image, then c([0,1]^k) is simply connected because it has a trivial fundamental group. Correct?
 

FAQ: Simple Connectedness of [0,1]^k

What is the definition of "Simple Connectedness"?

Simple connectedness is a topological property of a set that means the set is connected and every loop in the set can be continuously shrunk to a single point without leaving the set. In other words, there are no holes or gaps in the set that cannot be filled in without leaving the set.

How is the simple connectedness of [0,1]^k determined mathematically?

The simple connectedness of [0,1]^k can be determined using homotopy theory, which studies the continuous deformations of spaces. Specifically, the fundamental group of the set is used to determine if it is simply connected. If the fundamental group is trivial (i.e. contains only the identity element), then the set is simply connected.

What is the significance of [0,1]^k being simply connected?

Simply connected sets are important in topology because they are the simplest type of connected sets. They have no "holes" or "gaps" and can be continuously deformed into a single point. This property allows for simpler calculations and proofs in topology and other areas of mathematics.

What is the relationship between simple connectedness and path connectedness?

Path connectedness is a weaker condition than simple connectedness. A set is path connected if there exists a continuous path between any two points in the set. However, not all path connected sets are simply connected, as there may be loops that cannot be continuously shrunk to a single point. Therefore, simple connectedness implies path connectedness, but the reverse is not necessarily true.

Can a set be simply connected but not path connected?

No, a set cannot be simply connected but not path connected. This is because simply connectedness implies path connectedness. If a set is simply connected, then any loop can be continuously shrunk to a point, which means there must be a continuous path between any two points in the set. Therefore, a set must be both simply connected and path connected.

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