I was recently given this product space $X\times [0,1]$ with equivalence relation generated by $(x, 1) ~(y, 1)$. How to show the quotient space is connected? I just cannot do it Thanks all.
2025-01-12 23:57:51.1736726271
Question on quotient space
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It is actually path connected: Every elements in this quotient space are connected to $[X \times \{1\}]$ (This is a point in the quotient). To show this, just think of how every $(x, t) \in X\times [0,1]$ are connected to $(x, 1)$).
As path connected space is connected, you are done.