# Questions tagged [general-topology]

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

45,391 questions
Filter by
Sorted by
Tagged with
8 views

### Separability between $X$ and $\beta X$

Q) Is it true that if $\beta X$ is separable, then $X$ is also separable? I know if $X$ is locally compact, the result is true. Can someone help me with this?
19 views

### What's the difference between 1- Trace Topology, 2- Subspace Topology, 3- Relative Topology, and 4- Induced Topology?

I need help with some topology terms... So what is the difference between: 1- Trace Topology 2- Subspace Topology 3- Relative Topology 4- Induced Topology Or are they all the same thing?
42 views

27 views

### Showing a subset is open with respect to a topology

I have asked and read similar questions, but I am still somewhat confused on the notion of "a set being open with respect to some topology". My task is concerning the box topology and the ...
45 views

### Fundamental group of a bouquet of circles

Consider a bouquet of $n$ circles, centered at $(1,0), (2,0),...,(n,0)$ $$W_n= \bigcup_1^nC_r$$ with $$C_r:(x-r)^2+y^2=r^2$$ I want to compute the fundamental group of this space. $W_n$ is path ...
16 views

17 views

### Basic question on local compactness

Let $X$ be a topological space. Let $A \subset F \subset X$. Let $F$ be a subspace of $X$. Let $A$ be a subspace of $F$. Suppose that under this subspace topology inherited from $F$, $A$ forms a ...
11 views

### Not-continuos map with its graph closed.

Find an example of a map $f:(X,d_{X})\rightarrow (Y,d_{Y})$ between metric spaces such that $X$ is compact, $f$ is not continuous but $\Gamma(f):=\lbrace (x,f(x))\in X\times Y\mid x\in X\rbrace$ is ...
### Every continuous map $f:\mathbb{C}P(2) \to \mathbb{C}P(2)$ has a fixed point, without Lefschetz theorem.
I would like to know if there is a nice proof of the fact that every continuous map $f:\mathbb{C}P(2) \to \mathbb{C}P(2)$ has a fixed point, without use of the Lefschetz fixed point theorem.