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Topology: An Inquiry-Based Approach

Steven Schlicker

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Section Neighborhoods in Topological Spaces

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Recall that we defined a neighborhood of a point a in a metric space to be a subset of the space that contains an open ball centered at X. Every open ball is an open set, so we can extend the idea of neighborhood to topological spaces.
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Definition 12.6.

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Let (X,Ο„) be a topological space, and let a∈X. A subset N of X is a neighborhood of a if N contains an open set that contains a.
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Let’s look at some examples.
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In metric spaces, an open set was a neighborhood of each of its points. This is also true in topological spaces.
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Proof.

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Let (X,Ο„) be a topological space, and let O be a subset of X. First we demonstrate that if O is open, then O is a neighborhood of each of its points. Assume that O is an open set, and let a∈O. Then O contains the open set O that contains a, so O is a neighborhood of a.
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The reverse containment is the subject of the next activity.
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Activity 12.8.

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Let (X,Ο„) be a topological space. Let O be a subset of X. Assume O is a neighborhood of each of its points.
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(b)

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Let a∈O. Why must there exist an open set Oa such that a∈OaβŠ†O?
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