subspace topology
When you carve a subset out of a larger space, it should inherit a sense of openness from its surroundings rather than start from scratch. The subspace topology does exactly that: a piece of a subset counts as open if it is the part of some ambient open set that happens to lie inside the subset — open sets of the whole, intersected with the part.
Formally, if (X, T) is a topological space and A is a subset of X, the subspace topology on A consists of all sets of the form U ∩ A where U is open in X. This is genuinely a topology on A, and it makes the inclusion map of A into X continuous; it is the smallest topology doing so.
The crucial subtlety is that “open” and “closed” are relative to the ambient space you are in. A set can be open in A without being open in X — the half-open interval [0, 1) is open as a subset of [0, 2] (it equals (-1, 1) ∩ [0, 2]) even though it is not open in the full real line. So always ask “open in which space?” The good news is that compactness and connectedness of A do not depend on the ambient space, since they are intrinsic, but openness and closedness always do.
Let A = [0, 1] sit inside R. The set [0, 1/2) is open in A, because [0, 1/2) = (-1, 1/2) ∩ A and (-1, 1/2) is open in R — even though [0, 1/2) is not open in R. Openness is always a statement about a point relative to a chosen ambient space.
[0, 1/2) is open inside [0,1] but not inside R — openness depends on the ambient space.