Two topologies are called equivalent if any non-empty open set of the first topology contains a non-empty open set of the second topology and, conversely, every non-empty open set of the second topology contains non-empty open sets of the first topology.

2016年2月28日 · You do not need to consider the topologies 1 and 4 separately, this will already fall out ot considering 1 and 2 and 2 and 4. If we have strict inclusion there, then a fortiori we will have it for 1 and 4, no need for a separate proof. Similary with 2 and 5, e.g. What about 4 and 5? You just say "clear" but some argument seems in order.

2019年9月7日 · That are 3 more topologies (all homeomorphic to each other, if you already know that notion). One extra set that is a doubleton, so $\{a,b\}$, $\{a,c\}$ or $\{b,c\}$, again three extra topologies depending on which we have, all homeomorphic. That …

2015年3月7日 · However there are many useful/common topologies for the product of the topological spaces. The most common one is the product-topology. Another example is the box-topology, which is totally the same as the product-topology for finite products. Since you are new to topology. Topology is about defining the open sets by yourself.

2022年6月10日 · I have been studying topological vector spaces, and despite going over numerous resources, the definitions of weak and weak-* topologies have been causing me some confusion. I am having trouble visualizing and understanding these topologies.

Here, $\mathbb{R}_l$ is the lower limit topology on $\mathbb{R}$ and $\mathbb{R}_K$ is the K-topology on $\mathbb{R}$. I understand the proof that these topologies are strictly finer than $\mathbb{R}$, but I am at a loss to begin how to show they aren't comparable. This is …

Defining the class of sets for which the union of topologies is a topology [sets of subsets of the set of topologies on a set: the settening] 0 Partial order on the set of all topologies; minimal and maximal elements on this partial order.

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The discrete topology (all subsets open) is a topology for any set. So, take the set of all topologies containing the family of topologies (remember: the discrete one is here) and you will get a topology. It will be the smallest topology containing all topologies in the family. This is integral to the notion of weak topology.

2011年9月27日 · A union of topologies is a topology 0 Defining the class of sets for which the union of topologies is a topology [sets of subsets of the set of topologies on a set: the settening]