I am trying to solve the following question:
What familiar space is the quotient $Delta$-complex of a $2$ simplex$(v_0, v_1, v_2)$ obtained by identifying the edges $[v_0, v_1]$ and $[v_1, v_2],$ preserving the ordering of vertices?
But I am wondering why the word "quotient" is stated in the question and what will happen if ordering of vertices is not preserved? what will the shape we will get? I know that the shape we will get from the given question is the Mobius band.
In this context the quotient is the identification of the two edges. This is because quotient spaces in algebra are partitions of the original space, thus equivalence relations by the fundamental theorem of equivalence relations. The topological definition borrows from this idea so that topologically any partitioning is considered a quotient operation. Things like gluing the edges together is done by equivocating certain points on those edges, which is to say putting them in the same partition. You can also glue faces together, or points by putting them in the same partition.
So most of the points in the 2-simplex will be in their own partition but the points on the edges you're gluing together will be paired off in their partitions. Here you have two options, preserving the orientation or reversing it. You can think of this as adding a twist or no twist depending on the way the vertices are ordered. Just as you can make a Mobius band by introducing a twist to a strip of paper the other shape will be done without making that twist. Can you figure out what it is from here?