Define the terms connected and path connected for a topological space. If a topological space X is path connected, prove that it is connected.
Consider the following subsets of R2 :
I={(x,0):0≤x≤1},A={(0,y):21≤y≤1}, and Jn={(n−1,y):0≤y≤1} for n≥1
Let
X=A∪I∪n≥1⋃Jn
with the subspace (metric) topology. Prove that X is connected.
[You may assume that any interval in R (with the usual topology) is connected.]