**Metric Spaces First and Second Category MathReference**

Even nicer than rst-countable spaces are the second-countable, in which we can describe all the open sets in terms of a countable subcollection. De nition A space Xis second-countable if it has a countable basis for its topology,... If a metric space is Lindelof or separable then it is second countable. Subspaces and countability axioms: A subspace of a first-countable (second-countable) space is first-countable (second-countable).

**Proper metrics on locally compact groups and proper aï¬ƒne**

space is separable if it has a countable dense subset. Using the fact that any point in the closure of a set is the limit of a sequence in that set (yes?) it is easy to show that Q is dense in R,andsoR is separable. A discrete metric space is separable if and only if it is countable. My favourite example of a non-separable space is a hedgehog space of uncountable spinyness. One handy result is...Since a space with a locally countable weak-base is a generalization of a locally separable metric space, and since our purpose is to bring out proper- ties of locally separable metric spaces by means of that of the space with a

**B ASIC T OPOLOGY Pennsylvania State University**

If an y point of a topological space has a countable base of neighborhoods, then the space (or the topology) is called ?r st countable . E X A M P L E 1.1.10 . how to set up bully dog gt tuner space is separable if it has a countable dense subset. Using the fact that any point in the closure of a set is the limit of a sequence in that set (yes?) it is easy to show that Q is dense in R,andsoR is separable. A discrete metric space is separable if and only if it is countable. My favourite example of a non-separable space is a hedgehog space of uncountable spinyness. One handy result is. How to watch tv shows on ipad 2 for free

## How To Show A Metric Space Is Second Countable

### separable- LindelÃ¶f -- second countable for metric spaces

- Metric spaces SJTU
- Second-countable implies Lindelof Topospaces
- topology problems 2013 University of Arizona
- A NOTE ON SPACES WITH LOCALLY COUNTABLE univie.ac.at

## How To Show A Metric Space Is Second Countable

### property: for every neighbourhood Uof x, there exists j 1 such that U j U. We refer to B x as a local base at x. De nition. [Second countable] A topological space is called second countable if it admits a

- 4Countability axioms Denition 4.1. Let X be a topological space X is said to be rst countable if for any x 2 X , there is a countable basis for the neighborhoods of x. X is said to be second countable if there is a countable basis for the topology on X . Of course, second countable ) rst countable. First countability is a mild assump-tion, while having a countable basis gives much more
- Theorem 2 (Baire Category Theorem) A complete metric space cannot be expressed as a countable union of nowhere dense subsets. De?nition 3 A subset M of a topological space is said to be meagre (or ?rst category) if it can be expressed as the countable union of nowhere dense subsets. It is said to be nonmeagre (or second category) if it is not meagre. The Baire Category Theorem can be
- If a metric space is Lindelof or separable then it is second countable. Subspaces and countability axioms: A subspace of a first-countable (second-countable) space is first-countable (second-countable).
- In particular, we can prove that for a compact space K the space C p (K) is metrizable if and only if it is strongly dominated by a second countable space. We will also consider spaces C p ( X ) strongly dominated by second countable ones when X is a Tychonoff, not necessarily compact space.

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