Why is the "house with two rooms" is a 2-dimensional subspace of $\mathbb{R}^3$?

Question

We know that a plane is a two-dimensional subspace of $\mathbb{R}^3$. But in Hatcher's Algebraic Topology, the house with two rooms is a $2$-dimensional subspace of $\mathbb{R}^3$. Why is that? Isn't it supposed to be a $3$-dimensional subspace of $\mathbb{R}^3$?

What section/page are you on? – florence Aug 15 '16 at 05:51 — florence, Aug 15 '16 at 05:51
@florence Chapter 0 page 4. – user338393 Aug 15 '16 at 05:53 — user338393, Aug 15 '16 at 05:53

score 2 · Accepted Answer · answered Aug 15 '16 at 05:55

2

This is page 4 of Hatcher's book.

The "house" refers to the walls, ceiling and floor, not the air contained inside of it. Each of those locally looks like a plane, so the house itself locally (in most places) looks like a plane: 2-dimensional.

answered Aug 15 '16 at 05:55

user361430

36

Why is the "house with two rooms" is a 2-dimensional subspace of $\mathbb{R}^3$?

1 Answers1