What is the difference between the cone $$CX=X\times [0,1]/X\times \{0\}$$ and the open cone $$OC(X)=X\times [0,1)/X\times \{0\}?$$ I mean what is done by taking $[0,1)$ instead of $[0,1]$.
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$\begingroup$In the open cone we don't have the extra $X \times \{1\}$ that appears in the normal cone (note this end is not the end that is quotiented out).
For example if you take $CS^1$, the cone of the circle, you get a space homeomorphic to the closed disk (it includes the outer circle). However if we take $OC(S^1)$ we don't have this outer circle and the space is homeomorphic to the open disk as we do not have $S^1 \times \{1\}$ any more.
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