Subspace Topology of a Straight Line

[sammycaps]

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)?

 

[christoff]

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 $\mathbb{R}_l \times \mathbb{R}$, are indeed homeomorphic.

 

[sammycaps]

Alright, thanks!