Topology, Proofs, The word Complement

In summary, the conversation revolves around the concept of complement in a proof involving a closed interval. The individual is unsure of the meaning of complement and is looking for clarification or a definition. The suggested assumption is that the complement in this context would be (-oo, a) U (b, oo). They also mention a book that discusses the concept and suggest reading the first two pages of chapter 0 to gain a better understanding.
  • #1
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Topology, Proofs, The word "Complement"

Homework Statement


I have a proof to do in which they use the word "complement". I am not sure what it means by that withing the context of the question. There is no glossary to the book and there is no mention of complement before this question.


Homework Equations



Show that the complement of the closed interval [a,b] is an open set.

The Attempt at a Solution



I have not tried it yet because I do not know what they mean by complement. If you could point me to the right website where it talks about this or give a good definition I will then try and finish the proof from there.
 
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  • #2


The only assumption that I have is that the complement would be (-oo, a) U (b, oo). But still I am not sure if that is what they mean in this context.
 
  • #4


Thank you so much.
 

1. What is topology?

Topology is a branch of mathematics that studies the properties of geometric objects that remain unchanged when they are stretched, bent, or twisted. It focuses on the concept of continuity and the relationships between different points in a space.

2. What are some common applications of topology?

Topology has various applications in fields such as physics, engineering, computer science, and biology. It is used to analyze the shape of objects, understand networks and data structures, and study the behavior of complex systems.

3. What is a proof in mathematics?

In mathematics, a proof is a logical argument that shows that a statement or theorem is true. It involves using previously established axioms, definitions, and theorems to arrive at a conclusion that is supported by evidence and reasoning.

4. What is the word complement in mathematics?

In mathematics, the word complement refers to the set of elements in a particular space that are not included in a given set. For example, the complement of set A in set B would include all elements of set B that are not in set A.

5. How is topology related to the concept of complement?

In topology, the complement of a set refers to the set of all points outside of that set. This can be used to define topological spaces, where the complement of an open set is a closed set, and vice versa. The concept of complement is also important in understanding topological properties such as connectivity and compactness.

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