Question

Let E²

denote the real plane R²
endowed with the usual Euclidean topology. Consider the subspace Y=[0,1]×[0,1)={(x,y)∈R² :0≤x≤1,0≤y<1} of E²
Is Y compact?
1 - Yes
2 - No

Answer 1

Yes, the subspace Y=[0,1]×[0,1) of the real plane R² is compact.

Step-by-step explanation:

Yes, the subspace Y=[0,1]×[0,1) of the real plane R² is compact.

To show that Y is compact, we can use the Heine-Borel theorem, which states that a subspace in R² is compact if and only if it is closed and bounded.

In this case, Y is bounded because the values of x are between 0 and 1, and the values of y are between 0 and 1 (excluding 1). Y is also closed because it includes its boundary [0,1]×{0}.