Connectedness of subsets of connected closures

Thread starter mathnerd123
Start date Feb 15, 2010
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In summary, the connectedness of subsets of connected closures refers to the property of a subset of a connected closure being connected itself. It is determined by examining the topological structure of the subset and its closure. Yes, a subset of a connected closure can be disconnected, but it is an important concept in topology with applications in various fields. It is also closely related to the concept of continuity, as a continuous function preserves the connectedness of subsets.

Feb 15, 2010

mathnerd123

Can anyone help?

Given a set E in Rn is connected and E is a subset of A and A is a subset of E closure, and E closure is also connected, prove that A is connected.

Any help would be greatly appreciated!

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Feb 16, 2010

lanedance

Homework Helper

not 100%, but could you try and show the union of E a limit point of E is connected?

FAQ: Connectedness of subsets of connected closures

What is the definition of "connectedness of subsets of connected closures"?

The connectedness of subsets of connected closures refers to the property of a subset of a connected closure being connected itself. In other words, the subset cannot be divided into two or more disjoint nonempty open sets.

How is the connectedness of subsets of connected closures determined?

The connectedness of subsets of connected closures is determined by examining the topological structure of the subset and its closure. If the subset and its closure both have the same connectedness, then the subset is considered to have the connectedness of its closure.

Can a subset of a connected closure be disconnected?

Yes, a subset of a connected closure can be disconnected if it can be divided into two or more disjoint nonempty open sets. In this case, the subset does not have the connectedness of its closure.

What is the significance of studying the connectedness of subsets of connected closures?

The connectedness of subsets of connected closures is an important concept in topology, as it helps to understand the structure and behavior of topological spaces. It also has applications in many fields, such as computer science, physics, and engineering.

How is the connectedness of subsets of connected closures related to the concept of continuity?

The connectedness of subsets of connected closures is closely related to the concept of continuity in topology. A continuous function preserves the connectedness of subsets, meaning that if a subset is connected, its image under a continuous function will also be connected. This relationship allows us to use the concept of connectedness to prove the continuity of functions.