Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Distinguish a subset's geometry from its ambient topology.
- Justify the conclusion "The set is not open in R but is open in [0,2]" using the stated assumptions.
Before you start
Intervals and metric balls.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Is [0,1) open in R, and is it open in the subspace [0,2]?
Why this math matters
Distinguish a subset's geometry from its ambient topology. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- R has its usual metric topology.
- The subspace inherits that topology.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Test openness relative to the surrounding space
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Is [0,1) open in R, and is it open in the subspace [0,2]?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
In R, every ball around zero includes negative points
No positive-radius real ball around zero stays inside [0,1).
Work through the mathematics
[0,1)=[0,2]∩(−1,1)
The second factor is open in R.
Check the conclusion
The set is not open in R but is open in [0,2]
Subspace openness tests neighborhoods only after intersection with the ambient subspace.
The result
The set is not open in R but is open in [0,2]
Subspace openness tests neighborhoods only after intersection with the ambient subspace.
Common mistakes to catch
- Open and closed are not logical opposites.
- Always specify the ambient space.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Is [0,1] closed in R?
Show a hint
Examine its complement.
Reveal answer and explanation
Yes
The complement (−∞,0)∪(1,∞) is open.
Practice 2
Can a set be both open and closed?
Show a hint
Consider an entire space.
Reveal answer and explanation
Yes
The whole space and the empty set always have both properties.
Take the idea with you
Describe how boundary points change status when a sensor is restricted to an allowed interval.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Know when continuity guarantees an extreme value
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.