Subspace Topology of a Straight Line

In summary, when considering a vertical line L as a subspace of ℝl × ℝ, it inherits the standard topology of ℝ through its basis sets of the form {xo} × (c,d). This is due to the fact that L and ℝ are homeomorphic and thus, in the context of general topology, can be considered the same.
  • #1
sammycaps
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1. Hello, I'm reading through Munkres and I was doing this problem.

16.8) If L is a straight line in the plane, describe the topology L inherits as a subspace of ℝl × ℝ and as a subspace of ℝl × ℝl (where ℝl is the lower limit topology).

Homework Equations


The Attempt at a Solution



I've worked through it and seen many solutions online (they're all over). They essentially say that L, as a vertical line, inherits the standard topology on ℝ as a subspace of ℝl×ℝ, but some of the solutions jump from claiming that the basis sets in L are of the form {xo}×(c,d) for a vertical line L, which makes perfect sense, but then it jumps directly to saying that this implies that L inherits the standard topology, and I'm not sure I understand that (how can the topology of a 1-dimensional space be the topology on a 2-dimensional space?). Are the solutions maybe being informal, and instead mean that the topologies on L and R are "similar" (they are homeomorphic, but since homeomorphisms have not been introduced, we can't say this)?
 
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  • #2
The solutions are being informal. In the context of general topology, topological spaces (and indeed, subspaces under the induced topology, as these are topological spaces in their own right) which are homeomorphic, "are" the same. And your two spaces, the real line and a vertical line as a subspace of [itex] \mathbb{R}_l \times \mathbb{R} [/itex], are indeed homeomorphic.
 
  • #3
Alright, thanks!
 

1. What is a subspace topology?

A subspace topology is a mathematical concept that involves studying a subset of a topological space. It looks at how the open sets of the larger space behave when applied to the subset.

2. How is subspace topology applied to a straight line?

In the context of a straight line, subspace topology involves looking at how open sets on the line behave when applied to a subset of the line. This allows for the study of the properties and characteristics of the subset in relation to the larger line.

3. What is the difference between a subspace topology and a topology?

A topology is a general framework for studying the properties of a space, while a subspace topology is a more specific concept that focuses on studying a subset of a space. The subspace topology is derived from the larger topology, but it only considers the behavior of open sets on the subset.

4. How is the subspace topology of a straight line useful in mathematics?

The subspace topology of a straight line allows for the study of specific properties and characteristics of subsets on the line. It is also useful in understanding the relationship between a subset and the larger space, and in proving theorems and propositions about the subset.

5. Are there any real-world applications of subspace topology of a straight line?

Yes, there are various real-world applications of subspace topology of a straight line. For example, it can be used in physics to study the behavior of particles on a line, in economics to analyze the distribution of resources along a line, and in computer science to understand the behavior of data points on a line.

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