§ Nets from Munkres (TODO)

§ Directed set

A direct set is a partial order JJ which has "weak joins". That is, for every a,b∈Ja, b \in J, there exists a u∈Ju \in J such that a≤ua \leq u and b≤ub \leq u. It's not a join since we don't need uu to be UNIQUE.

§ Cofinal subset

A subset KK of a partial order JJ is said to be cofinal if loosely, ∀J≤∃K\forall J \leq \exists K. That is, for all j∈Jj \in J, there is a k∈Kk \in K such that j≤kj \leq k. So intuitively, KK is some sort of portion of JJ that leaves out a finite part of the bottom of JJ.

§ Cofinal subset is directed

§ Nets

Let XX be a topological space. a net is a function ff from a directed set JJ into XX. We usually write this as (xj)(x_j).

§ Net eventually in a subset AA

A net (xj)(x_j) is eventually in a subset AA if there exists an i∈Ii \in I such that for all j≥ij \geq i, xj∈Ax_j \in A. This is an ∃∀\exists \forall formula.

§ Net cofinally/frequently in a subset AA

The net x[:]x[:] is cofinally in a subset AA if the set {i∈I:xi∈A}\{ i \in I : x_i \in A \} is cofinal in II. This means that for all j∈Jj \in J, there exists a kk such that j≤kj \leq k and xk∈Ax_k \in A. This is a ∀∃\forall \exists formula. So intuitively, the net could "flirt" with the set AA, by exiting and entering with elements in AA.

§ Eventually in AA is stronger than frequently in AA

Let x[:]x[:] be a net that is eventually in AA. We will show that such a net is also frequently in AA. As the net is eventually in AA, there exists an e∈Ie \in I (for eventually), such that for all ii, e≤i  ⟹  x[i]∈Ae \leq i \implies x[i] \in A. Now, given an index ff (for frequently), we must establish an index which uu such that f≤u∧x[u]∈Af \leq u \land x[u] \in A. Pick uu as the upper bound of ee and ff which exists as the set II is directed. Hence, e≤u∧f≤ue \leq u \land f \leq u. We have that e≤u  ⟹  x[u]∈Ae \leq u \implies x[u] \in A. Thus we have an index uu such that f≤u∧x[u]∈Af \leq u \land x[u] \in A.

§ Not Cofinal/frequently in AA iff eventually in X−AX - A

§ Not Frequently in AA implies eventually in X−AX - A

Let the net be x[:]x[:] with index set II. Since we are not frequently in AA, this means that there is an index ff at which we are no longer frequent. That is, that there does not exist elements uu such that f≤u∧x[u]∈Af \leq u \land x[u] \in A. This means that for all elements uu such that f≤uf \leq u, we have x[u]∉Ax[u] \not \in A, or x[u]∈X−Ax[u] \in X - A. Hence, we can choose ff as the "eventual index", since all elements above ff are not in AA.

§ Eventually in X−AX - A implies not frequently in AA

Let the net be x[:]x[:] with index set II. Since the net is eventually in X−AX - A, this means that there is an index ee (for eventually) such that for all ii such that e≤ie \leq i we have x[i]∈X−Ax[i] \in X - A, or x[i]∉Ax[i] \not \in A. Thus, if we pick ee as the frequent index, we can have no index uu such that e≤u∧x[u]∈Ae \leq u \land x[u] \in A, since all indexes above ee are not in AA.

§ Convergence of a net

We say a net (xj)(x_j) converges to a limit l∈Xl\in X, written as (xj)→l(x_j) \rightarrow l iff for each neighbourhood UU of ll, there is a lower bound jU∈Jj_U \in J such that for all kk, jU≤k  ⟹  xk∈Uj_U \leq k \implies x_k \in U. That is, the image of the net after jUj_U lies in UU. In other words, the net (xj)(x_j) is eventually in every neighbourhood of ll. This is a ∀∃∀\forall \exists \forall formula (for all nbhd, exists cuttoff, for all terms above cutoff, we are in the nbhd)

§ Limit point of a net

We say that a point ll is a limit point of a net if xx is cofinally/frequently in every neighbourhood of AA. That is, for all neighbourhoods UU of AA, for all indexes j∈Jj \in J, there exists an index k[U,j]k[U, j] such that j≤k[U,j]∧x[j]∈Uj \leq k[U, j] \land x[j] \in U. This ia ∀∀∃\forall \forall \exists formula (for all nbhd, for all indexes, there exists a higher index that is in the nbhd).

§ Converge of a net with net as N\mathbb N

§ Product of nets

§ Convergenge of product nets iff component nets converge

§ Nets in Hausdorff spaces converge to at most one point

§ Point pp is in closure of AA iff net in AA converges to point pp

§ Function is continuous iff it preserves convergence of nets

§ Subnets

§ Subnets of a net converge

§ Accumulation point of a net

§ Subnets converge iff point is accumulation point

§ Compact implies every net has convergent subnet

§ Compact implies every net has convergent subnet

§ Universal Net

§ Every net has universal subnet

§ Universal net converges in compact space

§ pushforward of universal net is universal

§ Tychonoff's theorem

Let {Xα:α∈Λ}\{ X_\alpha : \alpha \in \Lambda \} is a collection of compact topological spaces. Let X≡∏α∈ΛXαX \equiv \prod_{\alpha \in \Lambda} X_\alpha be the product space. Let Φ:D→X\Phi: D \rightarrow X be a universal net for XX. For each λ∈Λ\lambda \in \Lambda, the push forward net πλ∘Φ:D→Xλ\pi_\lambda \circ \Phi: D \rightarrow X_\lambda is a universal net. Thus, it converges to some xλ∈Xλx_\lambda \in X_\lambda. Since products of nets converge iff their components converge, and here all the components converge, the original net also converges in XX. But this means that XX is compact as the universal net converges.