Neighborhood of a point

shashank dwivedi

Is a subset of a Neighborhood of a point is also a Neighborhood of the point? Justify

This question is being asked as an exercise after explaining the definition of Neighborhood of a point.

My deduction: At first go, it seems yes the statement is true. However, it says subset and not necessarily an interval. So, if I pick up any two elements randomly , then it will form a subset and it can happen that it doesn't contain any interval in which the point belongs. And hence, not every subset of a neighborhood of a point can be a neighborhood of the point.

Is my reasoning correct?

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If you say no, not every subset of a neighborhood is a neighborhood, then you need to provide a counterexample. Only a counterexample is a valid reasoning.

shashank dwivedi

Counterexample:

(-1,1) is a neighborhood of 0. However {0.5} being a singleton subset of this interval is not a neighborhood of the point.

Is this reasoning correct? I meant, that is this counterexample valid?

That works yes.