Further Analysis | Part IB, 2001

Define the terms connected and path connected for a topological space. If a topological space XX is path connected, prove that it is connected.

Consider the following subsets of R2\mathbb{R}^{2} :

I={(x,0):0x1},A={(0,y):12y1}, and Jn={(n1,y):0y1} for n1\begin{gathered} I=\{(x, 0): 0 \leq x \leq 1\}, \quad A=\left\{(0, y): \frac{1}{2} \leq y \leq 1\right\}, \text { and } \\ J_{n}=\left\{\left(n^{-1}, y\right): 0 \leq y \leq 1\right\} \quad \text { for } n \geq 1 \end{gathered}


X=AIn1JnX=A \cup I \cup \bigcup_{n \geq 1} J_{n}

with the subspace (metric) topology. Prove that XX is connected.

[You may assume that any interval in R\mathbb{R} (with the usual topology) is connected.]

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