Comparing Topologies on - Mathematics Stack Exchange
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.
real analysis - Equivalent Topologies - Mathematics Stack Exchange
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.
general topology - describing all possible topologies on a set ...
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.
Product of two topologies? - Mathematics Stack Exchange
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.
real analysis - On the definition of weak and weak-* topologies ...
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.
general topology - How to list all possible topologies
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 …
What exactly is the topology generated by the union of a family of ...
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.
Comparison of topologies - Mathematics Stack Exchange
Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Comparing certain topologies - Mathematics Stack Exchange
Consider the following topologies on $\mathbb{R}$ $\mathcal{T}_1=$ the standard topology $\mathcal{T}_2=$ the lower limit topology $\mathcal{T}_3=$ the topology having as basis all open rays $(- \
When is the union of topologies a 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]