# Continuous bijection from $[0,1]$ to $[0,0.5)\cup (0.5,1]$

Can we have a continuous bijection from $[0,1]$ to $[0,0.5)\cup (0.5,1]$?

• Can you find some topological property that one of the spaces has but the other hasn't? – Daniel Fischer Aug 21 '14 at 21:16
• Can you make some connection between the image of the set and the preimage? – IAmNoOne Aug 21 '14 at 21:17
• IVT – WimC Aug 21 '14 at 21:20

It is not possible because the set $[0,0.5) \cup (0.5, 1]$ is the union of two disjoint closed sets. So this space is not connected.
• You mean 'open' subsets of $[0,1]$, not closed sets, don't you? – Berci Aug 21 '14 at 22:21
• @Berci, I was referring to it as a subspace of $\mathbb{R}$. – IAmNoOne Aug 27 '14 at 21:16