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Suppose $X$ is a locally convex space and $Y$ is a subspace of the strong dual of $X$, is the induced topology on Y equivalent to the strong topology $b(Y,Y')$ on $Y$? … If this is not correct, then on what spaces does the strong topology coincide with the initial topology? …

with this topology is a Hausdorff topological field, and if $K$ is an ordered field then the order topology coincides with the valuation topology. … topology. …

EDIT (Harry): Since this question in its original form was poorly stated (asked about topology rather than graph theory), but we have a list of Topology books in the answers, I guess you should go ahead … EDIT (David): The original question was asking for places to learn topology with an eye towards applying it to computer science (artificial neural networks in particular) …

A dual of $E^{\star}$ given by the $\beta(E^{\star\star}, E^{\star})$ topology is usually denoted by $E^{\star\star}$ and it is called a double dual of $E$. … Fact 1 It is well-known that if $E$ is an (F)-space then the (initial) (F)-space topology coincides with strong topology $\beta(E, E^{\star})$. …

We define the order convergence topology, denoted by $\tau_o(P)$. … It is not hard to verify that this defines a topology. …

For a general topological vector space $X$ with chosen dual space $X',$ the "Mackey topology" is the strongest topology for which the continuous dual is $X'$. … My question is: is the canonical LF topology on $\mathcal{C}^{\infty}_{c}(\mathbb{R}^d)$ equal to the Mackey topology induced by the space of distributions? …

Now we define the order convergence topology, denoted by $\tau_o(P)$. … It is not hard to verify that this defines a topology. Question: Given any poset $(P,\leq)$, does $\tau_i(P)\subseteq \tau_o(P)$ hold? …

The pro-$W$ topology on a group $G$ is the unique group topology such that the set of normal subgroups $N$ with $G/N$ in $W$ is a fundamental system of neighborhoods of the identity. …

convex topology is barrelled) and similarly $\eta^{ub}$ as the associated ultrabornological topology. … a Banach space topology. …

Recall that for a Hausdorff locally convex space $X$ the Mackey topology $\tau (X^*,X)$ is the topology in its topological dual $X^*$ of uniform convergence on all weakly compact absolutely convex subsets …

Define a topology $\tau_1$ on $B(\mathcal H)$ by the family of seminorms $x\mapsto |Tr(xa)|,$ $a\in L^1(B(\mathcal H)).$ Here $B(\mathcal H)$ denotes the set of all bounded linear maps on $\mathcal H$ … Again define the $\sigma$-WOT topology $\tau_2$ on $B(\mathcal H)$ by pulling back the weak operator topology of $B(\mathcal H\otimes\ell_2)$ to $B(\mathcal H)$ via the map $x\mapsto x\otimes 1.$ How to …

In such case it is shown that the Alexandrov topology and the Manifolds topology deviate such that the manifold topology is strictly finer than the Alexandrov topology. … My question is that: The necessary and sufficient conditions under which two points that are distinguishable under the Manifold topology, are indistinguishable under the Alexandrov topology for a generic …

I recently went to a talk of Oleg Viro where he expressed his dissatisfaction with current foundations of differential topology parallel to what has been discussed here. … Like the language of topoi (and unlike 'tame topology'), it is a kind of topology 'without points' - a direct approach to 'shape'. ... appropriate for dealing with finite spaces... …

As someone more used to point-set topology, who is unfamiliar with the inner workings of lattice theory, I am looking to learn about the localic interpretation of topology, of which I only have a limited … What are some recent results in point-free topology that are unique to the subject, i.e., not translations of results from general topology into localic language? …

Is there an example of a metric space $(X,d)$ whose corresponding path metric, $d^\prime$ generates a strictly finer topology compared to the topology generated by $d$? …