A net is defined as a map $\Theta\to \mathbb{X}$ ($\theta\mapsto x_{\theta}$) where $\Theta$ is a directed set and $\mathbb{X}$ is some topological space. If $\Theta=\mathbb{N}$ then this definition coincides with the usual definition of a sequence. What would be an example of a net which is not a sequence? In particular for sequences we have that two different indices say $k\neq j$ map to the same element $x\in \mathbb{X}$ i.e. $x_k=x_j$. But the same index cannot be mapped to two different elements in $\mathbb{X}$ i.e. we cannot have $x_k=y_k$ where $x$ and $y$ are different. Is this the criteria which differentiates sequences from nets in general? Or do I get all this wrong?
[Math] A net vs a sequence
general-topologynetsreal-analysis
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The spaces characterized by the property that a subset is closed if and only if it is closed w.r.t. limits of transfinite sequences are called pseudoradial spaces.
I would like to give an example of a space that is not pseudoradial together with sketch of a proof. (I hope I am not doing something very easy in a too complicated way.) I would also like to mention a few references and some properties of these spaces.
Before presenting the example, one short remark. Pseudoradial spaces are represented by the convergence of net on well-ordered nets. Obviously, it is sufficient to take the nets on ordinals. We can go a little further - regular cardinals are sufficient. Indeed, if we have a cofinal subset of an ordinal, we can use this cofinal subset to get another convergent net.
Let us consider the following example. Each arrow in the picture bellow represents a convergent sequence. I.e., this is a topological space homeomorphic to $\{0\}\cup\{\frac1n;n\in\mathbb N\}$ taken as a subspace of real line. Equivalently, this is precisely the ordinal $\omega+1$ taken with the order topology.
We take all these sequences and identify some of the points as in the picture. (I.e., we make a quotient space of some of these spaces.) Let us call the resulting space $S_2$. Then we take the subspace of this space as shown in the picture. This subspace will be called $S_2^-$. (I've taken the notation $S_2$ and $S_2^-$ from this paper: Franklin S.P., Rajagopalan M., On subsequential spaces, Topology Appl. 35 (1990), 1-19. But you can notice that this space is very very similar to Arens-Fort space mentioned in Brian's answer.)
Now we want to show that $S_2^-$ is not pseudoradial.
Note that the space $S_2^-$ has only one non-isolated point. Let us call it $\omega$ . So we ask whether there is a transfinite sequence, consisting only of points different from $\omega$, which converges to $\omega$.
First, let us show that this is not possible for a regular cardinal $\alpha>\omega$. Suppose that $(x_\eta)_{\eta<\alpha}$ is an $\alpha$-sequence of points of $S_2^-\setminus\{\infty\}$, which converges to $\infty$. Let us denote $n_\eta$ the "column" to which $x_\eta$ belongs. In we use the notation the notation from the picture bellow $n_\eta$ is the first coordinate of ordered pair $x_\eta$.
We can see that $n_\eta$ converges to $\omega$. (E.g. by noticing that $(x,y)\mapsto x$ and $\omega\to\omega$ is a quotient map from $S_2^-$ to $\omega$ with order topology.)
Now this is not possible, since the we would be able to construct an increasing $\alpha$-sequence converging to $\omega$ and using this sequence we would be able to show that cofinality of $\alpha$ is $\omega$.
So the only possibility is to take a sequence in the usual sense, i.e., a sequence of length $\omega$. Perhaps with a little handwaving, but it is more-or-less clear that general situation is similar to the situation when the $n$-term of the sequence is in the $n$-th column. So we have a sequence $x_n=(n,y_n)$. Obviously $\{\omega\}\cup\bigcup\limits_{n\in\omega} \{n\}\times(y_n,\infty)$ is a neighborhood of the point $\omega$ containing no terms of this sequence.
Pseudoradial spaces were introduced by H. Herrlich. Quotienten geordneten Räume und Folgenkonvergenz. Fund. Math., 61:79–81, 1967; pdf. They were later studied by A.V. Arhangelskii and many others.
The class of pseudoradial spaces is closed under the formation of closed subspaces, quotients and topological sums. They are a coreflective subcategory of the category Top of all topological spaces. This means that for each topological space we have pseudoradial coreflection; a pseudoradial space which is, in some sense, close to this space. The pseudoradial coreflection is obtained simply by taking sets closed under limits of transfinite sequences as closed sets in a new topology on the same set. (E.g. the pseudoradial coreflection of $S_2^-$ is discrete.)
The same thing can be done with any class $\mathbb P$ of directed sets instead of ordinals. This is called $\mathbb P$-net spaces in P. J. Nyikos. Convergence in topology. (In M. Hušek and J. van Mill, editors, Recent Progress in General Topology, pages 537–570, Amsterdam 1992. North-Holland.) The properties of pseudoradial spaces which I mentioned in the preceding paragraph are true for $\mathbb P$-net spaces, too.
Interestingly, if we take the linearly ordered sets, we obtain the same class of spaces as from well-ordered sets, see James R. Boone: A note on linearly ordered net spaces. Pacific J. Math. Volume 98, Number 1 (1982), 25-35; link.
As far as I can say, the more usual definition of limit superior of a net is the one using limit of suprema of tails: $$\limsup x_d = \lim_{d\in D} \sup_{e\ge d} x_e = \inf_{d\in D} \sup_{e\ge d} x_e.$$
But you would get an equivalent definition, if you defined $\limsup x_d$ as the largest cluster point of the net. This definition corresponds (in a sense) to the definition with subsequential limits, since a real number is a cluster point of a net if and only if there is a subnet converging to this number.
I think that it is relatively easy to see that $\limsup x_d$ is a cluster point of the net $(x_d)_{d\in D}$.
To see that for every cluster point $x$ we have $x\le\limsup x_d$ it suffices to notice that, for any given $\varepsilon>0$ and $d\in D$, the interval $(x-\varepsilon,x+\varepsilon)$ must contain some element $x_e$ for $e\ge d$. Hence we get $$ \begin{align*} x-\varepsilon &\le \sup_{e\ge d} x_e\\ x-\varepsilon &\le \lim_{d\in D} \sup_{e\ge d} x_e. \end{align*}$$ and, since $\varepsilon>0$ is arbitrary, we get $$x\le \lim \sup_{e\ge d} x_e.$$
Thus the limit superior is indeed the maximal cluster point.
So the only thing missing is to show that cluster points are precisely the limits of subnets - this is a standard result, which you can find in many textbooks.
Some references for limit superior of a net are given in the Wikipedia article and in my answer here.
Perhaps some details given in my notes here can be useful, too. (The notes are still unfinished.) I should mention, that I pay more attention there to the notion of limit superior along a filter (you can find this in literature defined for filter base, which leads basically to the same thing). The limit superior of a net can be considered a special case, if we use the section filter; which is the filter generated by the base $\mathcal B(D)=\{D_a; a\in D\}$, where $D_a$ is the upper section $D_a=\{d\in D; d\ge a\}$.
Best Answer
An example that's important in proving that nets do the good things that they do: Say $X$ is a topological space and $p\in X$. Let $\Theta$ be the collection of all neighborhoods of $p$, ordered by reverse inclusion (so $V\ge W$ if $V\subset W$.
Recall that if $(x_\alpha)$ is a net in $X$ we say that $x_\alpha\to x$ if for every neighborhood $V$ of $x$ there exists $\beta$ such that $x_\alpha\in V$ for all $\alpha\ge\beta$. One of the reasons nets are useful is this:
If $f$ is continuous at $x$ and $x_\alpha\to x$ then it's trivial from the definitions that $f(x_\alpha)\to f(x)$. For the converse we need nets defined using that funny ordered set above.
Say $f$ is not continuous at $x$. So there exists $U\subset Y$ open with $f(x)\in U$ such that $f^{-1}(U)$ does not contain a neighborhood of $x$. Say $\Theta$ is the set of all neighborhoods of $x$, ordered by reverse inclusion. For every $V\in\Theta$ there exists $x_V\in V$ with $f(x_V)\notin U$. So $(x_V)$ is a net in $X$, and it's easy to verify that $x_V\to x$ but $f(x_V)\not\to f(x)$.
(The reason $\Theta$ was ordered by reverse inclusion was so we could show that $x_V\to x$: Say $W$ is a neighborhood of $x$. If $V\ge W$ then $x_V\in V\subset W$, hence $x_V\in W$ for every $V\ge W$.)