Hatcher-Algebraic-Topology · Problem 1.3.5

The Fundamental Group

Mathematics — Graph Theory · covering-spaces

Problem

Let

X

be the subspace of

\mathbb{R}^2

consisting of the four sides of the square

[0,1] \times [0,1]

together with the segments of the vertical lines

x = {}^1\!/_2, {}^1\!/_3, {}^1\!/_4, \cdots

inside the square. Show that for every covering space

\tilde{X} \to X

there is some neighborhood of the left edge of

X

that lifts homeomorphically to

\tilde{X}

. Deduce that

X

has no simply-connected covering space.