Topology Questions (studying for test)

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SUMMARY

The discussion centers on understanding the topology of the x-axis as a subspace of R² with the standard topology. Participants confirm that the topology for R² consists of open sets represented by little open circles, while the topology for any line in R², including the x-axis, is described by open intervals on the real line. The conversation emphasizes the importance of clarity in terminology and encourages starting new threads for distinct questions to facilitate better responses.

PREREQUISITES
  • Basic understanding of topology concepts, specifically subspaces.
  • Familiarity with R² and its standard topology.
  • Knowledge of open sets and intervals in real analysis.
  • Experience with mathematical notation and LaTeX formatting.
NEXT STEPS
  • Study the properties of subspaces in topology.
  • Learn about open sets in R² and their representations.
  • Explore the concept of one-dimensional manifolds in topology.
  • Review LaTeX formatting for mathematical expressions.
USEFUL FOR

Students preparing for topology exams, educators teaching topology concepts, and anyone interested in the mathematical foundations of subspace topology.

Damascus Road
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Greetings all,
I have a test upcoming next week and a lot of problems to solve. I asked in a previous thread whether or not I should just continue to adding to the thread and it seems the answer was yes. I appreciate any help! This is a tough subject.

Anyways, the first one I'm working on. I know it's simple but the wording is confusing me...

Let x = {(x,0) \in R^{2} | x \in R the axis in the plane. Describe the topology that inherits a subspace of R^{2} with the standard topology.

So, it's asking what topology the axis creates? Where the axis is the subspace, right?

(edit: I don't know why spaces within the tex seem to be ignored...)
 
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Greetings Damascus Road! :smile:

(type "\text{ Let }" :wink:)

the topology for R2 is little open circles …

so the topology for any line in R2 is … ? :wink

(btw, I would start a new thread each time)
 
Open intervals on the real line?
 
Yup! :biggrin:
 
But are those open intervals also circles?
 
(btw, I would start a new thread each time)


You don't think so many threads will annoy people? I don't want to make people mad and turn them off from helping :(
 
Damascus Road said:
But are those open intervals also circles?

they're one-dimensional circles! :biggrin:

(and no, you're far more likely to annoy people by dragging a thread out, and making them feel they have to keep replying to new questions that they weren't expecting! :wink:)
 
Ok thanks, I'll start another one :) :)
 

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