Let U be an open set of X and x∈ U. Which statement is correct?

U is a neighborhood of x, since x∈ U⊆ U with U open.

U is not a neighborhood of x because neighborhoods cannot be open.

U is a neighborhood of x only if U=X.

U is a neighborhood of x only if U is closed.

Exercise · ZW-NEIGHBORHOOD-007 · ★★☆ Intermediate

Let \(U\) be an open set of \(X\) and \(x\in U\). Which statement is correct?

Choose one option.

Show answer

A. \(U\) is a neighborhood of \(x\), since \(x\in U\subseteq U\) with \(U\) open.

Explanation

Taking the open set \(U\) itself in the definition gives \(x\in U\subseteq U\); every open set is a neighborhood of each of its points.

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Review the concept: Neighborhood of a point

A neighborhood of a point \(x\) in a topological space \((X,\mathcal{T})\) is a set \(N\subseteq X\) that contains some open set \(U\in\mathcal{T}\) to which \(x\) belongs.

\[N\ \text{neighborhood of }x\iff \exists\,U\in\mathcal{T}:\ x\in U\subseteq N\]

Let \(U\) be an open set of \(X\) and \(x\in U\). Which statement is correct?

Explanation

Review the concept: Neighborhood of a point

More practice problems: neighborhood of a point