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Since when I started studying general topology there is something concearning neighborhoods which baffles me.

Given a topological space $(X, \mathscr{T})$ and $p\in X$ we say $U\subseteq X$ is a neighborhood of $x$ if there is $W\in\mathscr{T}$ such that $x\in W\subseteq U$.

Naturally, every open set $W\in\mathscr{T}$ containing $x$ is a neighborhood of $x$ which we call open neighborhood of $x$.

There are several topological properties (not in the sense of topological invariance) which can be described in terms of neighborhoods like continuity, haudorffness, etc.

Of course if a given property holds for a neighborhood it also holds for an open neighborhood.

Can anyone give me some examples where some property holds for open neighborhoods but it fails for arbitrary ones?

Continuity and Hausdorfness won't work for we might interchange neighborhoods and open neighborhoods and we get equivalent definitions.

Thanks.

PtF
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3 Answers3

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A space $\langle X,\tau\rangle$ is $T_0$ if for any two distinct points $x,y\in X$ there is an open set $U$ such that either $x\in U$ and $y\notin U$, or $y\in U$ and $x\notin U$. In other words, at least one of the points has an open nbhd that does not contain the other. This is a much weaker property even than $T_1$: the Sierpiński space is $T_0$ but not $T_1$ (and hence certainly not Hausdorff).

Now suppose that $X$ has the property that if $x$ and $y$ are distinct points of $X$, then at least one of $x$ and $y$ has a closed nbhd that does not contain the other. Then $X$ is actually Hausdorff: if $H$ is a closed nbhd of $x$, say, that does not contain $y$, then $\operatorname{int}H$ and $X\setminus H$ are disjoint open nbhds of $x$ and $y$, respectively.

The most common definition of local compactness is that $X$ is locally compact if each point of $X$ has a compact nbhd. This is plainly not equivalent to saying that each point of $X$ has a compact open nbhd: $\Bbb R$ with the usual topology is locally compact, but no point of $\Bbb R$ has a compact open nbhd.

Brian M. Scott
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Separation by open neighborhoods and by closed neighborhoods are not equivalent.

A topological space is Hausdorff, (or $T_2$) if any two distict points can be separated by open neighborhoods.

An Urysohn space , or $T_{2\frac{1}{2}}$ space, is a topological space where any two distict points can be separated by closed neighborhoods.

We can prove that $T_{2\frac{1}{2}} \Rightarrow T_2$, but there exists topological spaces that are $T_2$ but not $T_{2\frac{1}{2}}$.

For an exemple see: Example of Hausdorff space $X$ s.t. $C_b(X)$ does not separate points?.

Community
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Emilio Novati
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For the concept of local connectedness at a point $x$, there is a distinction between having a neighborhood base of open connected sets (locally connected at $x$) and having a neighborhood base of connected sets at $x$ (connected im kleinen at $x$). These definitions are not equivalent.

By the way, the corresponding definitions on the space $X$ are equivalent: $X$ is locally connected iff $X$ is connected im kleinen.

b yen
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