The meaning of different in Munkres' Topology

In summary, "different" in this context means that the four topologies discussed are not equal and may or may not be comparable. The problem is asking to show that these topologies are distinct and have different properties.
  • #1
madsmh
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The meaning of "different" in Munkres' Topology

Hi, I'm working on problem 20.8(b) (page 127f) in Munkres' "Topology", the problem is to show that four topologies are "different". Does different in this context mean that they are unequal - in which case one can contain the other, or non-comparable - in which case none may contain the other?

Thanks in advance!

. Mads
 
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  • #2


Well, the inclusion in a) still applies in the subspace, so it's definitely not that they cannot be comparable. Different mean different as sets here.
 
  • #3
Hi Mads,

In this context, "different" means that the four topologies are not equal. This means that one may contain the other, or they may be non-comparable. The problem is asking you to show that these topologies are distinct, meaning they have different properties and cannot be considered the same topology.

Hope this helps clarify things! Let me know if you have any other questions. Good luck with the problem!
 

1. What is the definition of "different" in Munkres' Topology?

In Munkres' Topology, "different" refers to two sets or points that are distinct or separate from each other. This means that they do not share any elements or have any points in common.

2. How does Munkres' Topology define "different" from other mathematical definitions?

Munkres' Topology defines "different" in a similar manner to other mathematical definitions, such as the concept of disjointness in set theory. However, in topology, the emphasis is on the spatial or geometric aspect of being different, rather than just the set-theoretic aspect.

3. Can two sets or points be different in Munkres' Topology if they have some points in common?

No, in Munkres' Topology, two sets or points are considered different if they have no points in common. If they have some points in common, they are considered to be the same set or point.

4. How does the concept of "different" relate to the concept of "open" in Munkres' Topology?

In Munkres' Topology, two sets or points are considered different if they are not contained in the same open set. This is because in topology, open sets are used to define the notion of proximity or separation between points.

5. Are there any exceptions to the definition of "different" in Munkres' Topology?

Yes, there are some exceptions to the definition of "different" in Munkres' Topology. For example, in the case of topological spaces with multiple connected components, two points in different connected components may still be considered different, even if they have some points in common.

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