# Questions tagged [nets]

A net is a generalization of a sequence where a directed set is used as the index set instead of positive integers. Convergence of nets can be defined in a similar way as convergence of sequences. Convergent nets in a topological space uniquely determine its topology.

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### If $Σ$ is endowed with the initial topology $\cal T_ Φ$ corresponding to the collection $Φ$ then $σ_i⟶σ$ iff $φ(σ_i)⟶φ(σ)$ for any $φ∈Φ$.

When I started to study topology of compact convergence it seemed to me that there was something not said explicitly that really help to understand better this topology so that I observed that for any ...
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### Cofinality of a set

Let $(I, \leq)$ be a directed set. Suppose that $I$ has uncountable cofinality and that one can decompose $I$ as a countable union $I = \bigcup_{n = 0}^\infty I_n$, is it true that at least one of the ...
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### Lower semi-continuous function attains minimum on a compact set [duplicate]

I'm trying to use net to prove this well-known result. Could you have a check on my attempt? Let $E$ be a compact topological space and $f:E \to \mathbb R$ lower semi-continuous. Then $f$ attains the ...
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### Proof of Hausdorffness of sequentially Hausdorff space under its sequential topology

Under "Topology of sequentially open sets" section of the Wikipedia page Sequential Space, there is a claim which says any sequentially Hausdorff(i.e. every convergent sequence has a unique ...
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